mathematical biology - background, key issues · 2014-04-15 · overview of week ahead single...
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Overview of week aheadSingle Population Models
Coupled population modelsSummary
Mathematical Biology - Background, Key Issues
Stuart Townley
University of Exeter, UK
March 17, 2014
Stuart Townley Math Biol - Basics 1/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
Lecture 1: Mathematical biology for one and two dimensionalmodels
Lecture 2: Population Projection Models - A Lecture by“Crowd Sourcing”Built into this topic is a group research project - Presentationson Friday
Lecture 3: A Feedback Control Approach to NonlinearPopulation Dynamics
Lecture 4: Diffusion Driven Instability and links to SwitchedSystems
Lecture 5: Further and future topics in “Systems Theory forMathematical Biology”
Stuart Townley Math Biol - Basics 2/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
Lecture 1: Mathematical biology for one and two dimensionalmodels
Lecture 2: Population Projection Models - A Lecture by“Crowd Sourcing”Built into this topic is a group research project - Presentationson Friday
Lecture 3: A Feedback Control Approach to NonlinearPopulation Dynamics
Lecture 4: Diffusion Driven Instability and links to SwitchedSystems
Lecture 5: Further and future topics in “Systems Theory forMathematical Biology”
Stuart Townley Math Biol - Basics 2/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
Lecture 1: Mathematical biology for one and two dimensionalmodels
Lecture 2: Population Projection Models - A Lecture by“Crowd Sourcing”Built into this topic is a group research project - Presentationson Friday
Lecture 3: A Feedback Control Approach to NonlinearPopulation Dynamics
Lecture 4: Diffusion Driven Instability and links to SwitchedSystems
Lecture 5: Further and future topics in “Systems Theory forMathematical Biology”
Stuart Townley Math Biol - Basics 2/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
Lecture 1: Mathematical biology for one and two dimensionalmodels
Lecture 2: Population Projection Models - A Lecture by“Crowd Sourcing”Built into this topic is a group research project - Presentationson Friday
Lecture 3: A Feedback Control Approach to NonlinearPopulation Dynamics
Lecture 4: Diffusion Driven Instability and links to SwitchedSystems
Lecture 5: Further and future topics in “Systems Theory forMathematical Biology”
Stuart Townley Math Biol - Basics 2/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
Lecture 1: Mathematical biology for one and two dimensionalmodels
Lecture 2: Population Projection Models - A Lecture by“Crowd Sourcing”Built into this topic is a group research project - Presentationson Friday
Lecture 3: A Feedback Control Approach to NonlinearPopulation Dynamics
Lecture 4: Diffusion Driven Instability and links to SwitchedSystems
Lecture 5: Further and future topics in “Systems Theory forMathematical Biology”
Stuart Townley Math Biol - Basics 2/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Birth, growth, logistic growth
Hysteresis effect
Harvesting
Cob-web models
Chaos
Stuart Townley Math Biol - Basics 3/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Let N(t) denote the total population of a species at time t.Then
dN
dt= “Births− deaths + migration”.
Malthus (1798) proposed a simplified model for the right handside with
birth and death proportional to N
dN
dt= αN − β N, for constants α (fecundity), β (mortality).
Stuart Townley Math Biol - Basics 4/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Let N(t) denote the total population of a species at time t.Then
dN
dt= “Births− deaths + migration”.
Malthus (1798) proposed a simplified model for the right handside with
birth and death proportional to N
dN
dt= αN − β N, for constants α (fecundity), β (mortality).
Stuart Townley Math Biol - Basics 4/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Let N(t) denote the total population of a species at time t.Then
dN
dt= “Births− deaths + migration”.
Malthus (1798) proposed a simplified model for the right handside with
birth and death proportional to N
dN
dt= αN − β N, for constants α (fecundity), β (mortality).
Stuart Townley Math Biol - Basics 4/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Separating variables
dN
αN − βN= dt
solving for N by integrating left and right hand sides yields:
N(t) = N0e(α−β)t, for initial N0.
ThenN(t) →∞ if α > β
→ 0 if α < β .
We will return to this simplified view of birth and deathprocesses in a context of multiple age- or stage-structuredpopulations involving also growth between stages (See MBLecture 2).
Stuart Townley Math Biol - Basics 5/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Separating variables
dN
αN − βN= dt
solving for N by integrating left and right hand sides yields:
N(t) = N0e(α−β)t, for initial N0.
ThenN(t) →∞ if α > β
→ 0 if α < β .
We will return to this simplified view of birth and deathprocesses in a context of multiple age- or stage-structuredpopulations involving also growth between stages (See MBLecture 2).
Stuart Townley Math Biol - Basics 5/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Separating variables
dN
αN − βN= dt
solving for N by integrating left and right hand sides yields:
N(t) = N0e(α−β)t, for initial N0.
ThenN(t) →∞ if α > β
→ 0 if α < β .
We will return to this simplified view of birth and deathprocesses in a context of multiple age- or stage-structuredpopulations involving also growth between stages (See MBLecture 2).
Stuart Townley Math Biol - Basics 5/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Separating variables
dN
αN − βN= dt
solving for N by integrating left and right hand sides yields:
N(t) = N0e(α−β)t, for initial N0.
ThenN(t) →∞ if α > β
→ 0 if α < β .
We will return to this simplified view of birth and deathprocesses in a context of multiple age- or stage-structuredpopulations involving also growth between stages (See MBLecture 2).
Stuart Townley Math Biol - Basics 5/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
In this single stage/species model we question thissimplification and introduce self-limited growth, Verhulst(1836):
dN
dt= rN
(1− N
K
), r,K > 0 (1)
This is called logistic growth; r is called the linear birth rateand K is the carrying capacity.
Separation of variables in (1) and integration yields:
N(t) = N0Kert
K+N0(ert−1)→ K as t→∞
Stuart Townley Math Biol - Basics 6/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
In this single stage/species model we question thissimplification and introduce self-limited growth, Verhulst(1836):
dN
dt= rN
(1− N
K
), r,K > 0 (1)
This is called logistic growth; r is called the linear birth rateand K is the carrying capacity.
Separation of variables in (1) and integration yields:
N(t) = N0Kert
K+N0(ert−1)→ K as t→∞
Stuart Townley Math Biol - Basics 6/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
In this single stage/species model we question thissimplification and introduce self-limited growth, Verhulst(1836):
dN
dt= rN
(1− N
K
), r,K > 0 (1)
This is called logistic growth; r is called the linear birth rateand K is the carrying capacity.
Separation of variables in (1) and integration yields:
N(t) = N0Kert
K+N0(ert−1)→ K as t→∞
Stuart Townley Math Biol - Basics 6/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
The steady states of (1) are N = 0 and N = K. ClearlyN = 0 is unstable, while N = K is exponentially stable.
0 5 10 150
0.5
1
1.5
2
2.5
3
t
N(t)
Label the curves accordingto their respective initialconditions
Stuart Townley Math Biol - Basics 7/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Single species models with generalised population dynamics
A natural extension of the logistic growth model (1) is
dN
dt= f(N) (2)
with nonlinear birth-death-migration dynamics f(N).
Equilibrium populations N∗ are determined from f(N∗) = 0.
We can determine whether N∗ is stable by considering smallperturbations around N∗ so that N(t) = N∗ + n(t) with n(t)small.
Stuart Townley Math Biol - Basics 8/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Single species models with generalised population dynamics
A natural extension of the logistic growth model (1) is
dN
dt= f(N) (2)
with nonlinear birth-death-migration dynamics f(N).
Equilibrium populations N∗ are determined from f(N∗) = 0.
We can determine whether N∗ is stable by considering smallperturbations around N∗ so that N(t) = N∗ + n(t) with n(t)small.
Stuart Townley Math Biol - Basics 8/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Single species models with generalised population dynamics
A natural extension of the logistic growth model (1) is
dN
dt= f(N) (2)
with nonlinear birth-death-migration dynamics f(N).
Equilibrium populations N∗ are determined from f(N∗) = 0.
We can determine whether N∗ is stable by considering smallperturbations around N∗ so that N(t) = N∗ + n(t) with n(t)small.
Stuart Townley Math Biol - Basics 8/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
For N∗ an equilibrium, f(N∗) = 0. Therefore theperturbation n(t) satisfies
n ≈ f ′(N∗)n
so thatn(t) ≈ n0e[f
′(N∗)t]
and, analogously with the linear birth-death model,
n(t)→∞ if f ′(N∗) > 0 and n(t)→ 0 if f ′(N∗) < 0 .
So
N∗ is unstable if f ′(N∗) > 0 and
exponentially stable if f ′(N∗) < 0.
Stuart Townley Math Biol - Basics 9/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
For N∗ an equilibrium, f(N∗) = 0. Therefore theperturbation n(t) satisfies
n ≈ f ′(N∗)n
so thatn(t) ≈ n0e[f
′(N∗)t]
and, analogously with the linear birth-death model,
n(t)→∞ if f ′(N∗) > 0 and n(t)→ 0 if f ′(N∗) < 0 .
So
N∗ is unstable if f ′(N∗) > 0 and
exponentially stable if f ′(N∗) < 0.
Stuart Townley Math Biol - Basics 9/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
For N∗ an equilibrium, f(N∗) = 0. Therefore theperturbation n(t) satisfies
n ≈ f ′(N∗)n
so thatn(t) ≈ n0e[f
′(N∗)t]
and, analogously with the linear birth-death model,
n(t)→∞ if f ′(N∗) > 0 and n(t)→ 0 if f ′(N∗) < 0 .
So
N∗ is unstable if f ′(N∗) > 0 and
exponentially stable if f ′(N∗) < 0.
Stuart Townley Math Biol - Basics 9/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Define the time-response of the equilibrium population N∗ tobe the interval of time in which the perturbed population n ofthe unstable, resp. stable, linearised dynamics, increases, resp.decreases, by a factor e.
So,
time-response =1
|f ′(N∗)|.
Typically, f(N) = 0 has several solutions, thus the populationdynamics exhibit multiple equilibria. A sketch of the graph off(N) readily determines their stability type simply byinspecting the sign of f ′(N∗).
Stuart Townley Math Biol - Basics 10/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Define the time-response of the equilibrium population N∗ tobe the interval of time in which the perturbed population n ofthe unstable, resp. stable, linearised dynamics, increases, resp.decreases, by a factor e.
So,
time-response =1
|f ′(N∗)|.
Typically, f(N) = 0 has several solutions, thus the populationdynamics exhibit multiple equilibria. A sketch of the graph off(N) readily determines their stability type simply byinspecting the sign of f ′(N∗).
Stuart Townley Math Biol - Basics 10/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Define the time-response of the equilibrium population N∗ tobe the interval of time in which the perturbed population n ofthe unstable, resp. stable, linearised dynamics, increases, resp.decreases, by a factor e.
So,
time-response =1
|f ′(N∗)|.
Typically, f(N) = 0 has several solutions, thus the populationdynamics exhibit multiple equilibria. A sketch of the graph off(N) readily determines their stability type simply byinspecting the sign of f ′(N∗).
Stuart Townley Math Biol - Basics 10/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Modelling an Outbreak of Spruce Budworm
A major problem in Canadian Forests is caused by the SpruceBudworm devouring the foliage of fir trees.
Let N(t) be the number of budworms. A plausible model forN(t) is:
dN
dt= rBN
(1− N
KB
)− p(N) (3)
Here rB is the birth rate,KB is the carrying capacity,p(N) is a nonlinear function modelling predation by birds.Typical characteristics of p(N) are depicted in the graphbelow:
One possibility for p(N) is p(N) = BN2
A2+N2 .
Stuart Townley Math Biol - Basics 11/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Modelling an Outbreak of Spruce Budworm
A major problem in Canadian Forests is caused by the SpruceBudworm devouring the foliage of fir trees.
Let N(t) be the number of budworms. A plausible model forN(t) is:
dN
dt= rBN
(1− N
KB
)− p(N) (3)
Here rB is the birth rate,KB is the carrying capacity,p(N) is a nonlinear function modelling predation by birds.Typical characteristics of p(N) are depicted in the graphbelow:
One possibility for p(N) is p(N) = BN2
A2+N2 .
Stuart Townley Math Biol - Basics 11/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Modelling an Outbreak of Spruce Budworm
A major problem in Canadian Forests is caused by the SpruceBudworm devouring the foliage of fir trees.
Let N(t) be the number of budworms. A plausible model forN(t) is:
dN
dt= rBN
(1− N
KB
)− p(N) (3)
Here rB is the birth rate,KB is the carrying capacity,p(N) is a nonlinear function modelling predation by birds.Typical characteristics of p(N) are depicted in the graphbelow:
One possibility for p(N) is p(N) = BN2
A2+N2 .
Stuart Townley Math Biol - Basics 11/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
N
pred
atio
n by
bird
s p(
N)
saturation level
Stuart Townley Math Biol - Basics 12/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
To start our analysis we first non-dimensionalise, so as toreduce the number of crucial parameters. In fact with
u =N
Aand τ =
Bt
A
we end up with
du
dτ=ArBB
u
(1− A
KBu
)− u2
1 + u2.
This suggests a rescaling of parameters
q =KB
A, r =
ArBB
so yielding a two-parameter model:
du
dτ= ru
(1− u
q
)− u2
1 + u2:= f(u; r; q) . (4)
Stuart Townley Math Biol - Basics 13/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
To start our analysis we first non-dimensionalise, so as toreduce the number of crucial parameters. In fact with
u =N
Aand τ =
Bt
A
we end up with
du
dτ=ArBB
u
(1− A
KBu
)− u2
1 + u2.
This suggests a rescaling of parameters
q =KB
A, r =
ArBB
so yielding a two-parameter model:
du
dτ= ru
(1− u
q
)− u2
1 + u2:= f(u; r; q) . (4)
Stuart Townley Math Biol - Basics 13/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
To start our analysis we first non-dimensionalise, so as toreduce the number of crucial parameters. In fact with
u =N
Aand τ =
Bt
A
we end up with
du
dτ=ArBB
u
(1− A
KBu
)− u2
1 + u2.
This suggests a rescaling of parameters
q =KB
A, r =
ArBB
so yielding a two-parameter model:
du
dτ= ru
(1− u
q
)− u2
1 + u2:= f(u; r; q) . (4)
Stuart Townley Math Biol - Basics 13/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
The equilibria of (4) are given by f(u; r; q) = 0, i.e.
ru
(1− u
q
)=
u2
1 + u2.
Clearly u = 0 is always a solution.The non-zero equilibria are given by:
r
(1− u
q
)=
u
1 + u2. (5)
How many non-zero equilibria will depend on r and q. This isbest analysed graphically by plotting both the left and righthand sides of (5) on the same graph. This is shown below.
Stuart Townley Math Biol - Basics 14/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
The equilibria of (4) are given by f(u; r; q) = 0, i.e.
ru
(1− u
q
)=
u2
1 + u2.
Clearly u = 0 is always a solution.The non-zero equilibria are given by:
r
(1− u
q
)=
u
1 + u2. (5)
How many non-zero equilibria will depend on r and q. This isbest analysed graphically by plotting both the left and righthand sides of (5) on the same graph. This is shown below.
Stuart Townley Math Biol - Basics 14/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
The equilibria of (4) are given by f(u; r; q) = 0, i.e.
ru
(1− u
q
)=
u2
1 + u2.
Clearly u = 0 is always a solution.The non-zero equilibria are given by:
r
(1− u
q
)=
u
1 + u2. (5)
How many non-zero equilibria will depend on r and q. This isbest analysed graphically by plotting both the left and righthand sides of (5) on the same graph. This is shown below.
Stuart Townley Math Biol - Basics 14/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
0 1 2 3 4 5 6 7 8 9 100
0.125
0.25
0.375
0.5
u
One non−zero Eqm, r=a
Two non−zero Eqm, r=b, d
Three non−zero Eqm, r=c
r=br=c
r=d
r=a
Stuart Townley Math Biol - Basics 15/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
We see from this sketch that if q is large enough, then
r < d there is one small stable non-zero equilibrium
r = d there are two non-zero equilibria
d < r < b there are three non-zero equilibria
r = b there are two non-zero equilibria
r > b there is one large stable non-zero equilibrium
As we change the parameter, we get differing numbers ofequilibria and in this example this induces tipping points andhysteresis effect.
Stuart Townley Math Biol - Basics 16/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
We see from this sketch that if q is large enough, then
r < d there is one small stable non-zero equilibrium
r = d there are two non-zero equilibria
d < r < b there are three non-zero equilibria
r = b there are two non-zero equilibria
r > b there is one large stable non-zero equilibrium
As we change the parameter, we get differing numbers ofequilibria and in this example this induces tipping points andhysteresis effect.
Stuart Townley Math Biol - Basics 16/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
non−
zero
equ
ilibr
ia
d < r < b, three equilibriar < d,one eqm
STABLE
UNSTABLE
STABLE
r > bone eqm
r=d r=b
Stuart Townley Math Biol - Basics 17/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Harvesting a single natural population
Harvested populations are everywhere - bees, cod (and otherfish species), whale, rabbits, etc.
Harvesting is desirable - as a food resource - and necessary -in order to control species numbers.
Typically we look to maximise sustainable yield whilstminimising harvesting effort.
Stuart Townley Math Biol - Basics 18/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Harvesting a single natural population
Harvested populations are everywhere - bees, cod (and otherfish species), whale, rabbits, etc.
Harvesting is desirable - as a food resource - and necessary -in order to control species numbers.
Typically we look to maximise sustainable yield whilstminimising harvesting effort.
Stuart Townley Math Biol - Basics 18/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Harvesting a single natural population
Harvested populations are everywhere - bees, cod (and otherfish species), whale, rabbits, etc.
Harvesting is desirable - as a food resource - and necessary -in order to control species numbers.
Typically we look to maximise sustainable yield whilstminimising harvesting effort.
Stuart Townley Math Biol - Basics 18/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
To illustrate harvesting, consider a model system which,without harvesting, obeys a logistics model with birth rate rand carrying capacity K.
Add to this a harvesting term, hN , proportional to population:
dN
dt= rN
(1− N
K
)− hN ≡ f(N) . (6)
The steady states are
N = 0, Nh = K
(1− h
r
)> 0 if r > h .
The steady state Nh gives a yield hNh = hK(1− h
r
).
Maximum steady state yield is achieved at h = r/2 (bystandard calculus) giving
maximum yield Ymax = Y (h)|h=r/2 = rK4
at Nh|Ymax = 12K
(7)
Stuart Townley Math Biol - Basics 19/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
To illustrate harvesting, consider a model system which,without harvesting, obeys a logistics model with birth rate rand carrying capacity K.Add to this a harvesting term, hN , proportional to population:
dN
dt= rN
(1− N
K
)− hN ≡ f(N) . (6)
The steady states are
N = 0, Nh = K
(1− h
r
)> 0 if r > h .
The steady state Nh gives a yield hNh = hK(1− h
r
).
Maximum steady state yield is achieved at h = r/2 (bystandard calculus) giving
maximum yield Ymax = Y (h)|h=r/2 = rK4
at Nh|Ymax = 12K
(7)
Stuart Townley Math Biol - Basics 19/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
To illustrate harvesting, consider a model system which,without harvesting, obeys a logistics model with birth rate rand carrying capacity K.Add to this a harvesting term, hN , proportional to population:
dN
dt= rN
(1− N
K
)− hN ≡ f(N) . (6)
The steady states are
N = 0, Nh = K
(1− h
r
)> 0 if r > h .
The steady state Nh gives a yield hNh = hK(1− h
r
).
Maximum steady state yield is achieved at h = r/2 (bystandard calculus) giving
maximum yield Ymax = Y (h)|h=r/2 = rK4
at Nh|Ymax = 12K
(7)
Stuart Townley Math Biol - Basics 19/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
To illustrate harvesting, consider a model system which,without harvesting, obeys a logistics model with birth rate rand carrying capacity K.Add to this a harvesting term, hN , proportional to population:
dN
dt= rN
(1− N
K
)− hN ≡ f(N) . (6)
The steady states are
N = 0, Nh = K
(1− h
r
)> 0 if r > h .
The steady state Nh gives a yield hNh = hK(1− h
r
).
Maximum steady state yield is achieved at h = r/2 (bystandard calculus) giving
maximum yield Ymax = Y (h)|h=r/2 = rK4
at Nh|Ymax = 12K
(7)
Stuart Townley Math Biol - Basics 19/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Does a dynamical analysis give us anything different?
Linearising (6) about the non-zero equilibrium Nh gives:
dn
dt= (f ′(Nh))n = (h− r)n .
Thus the non-zero equilibrium is linearly stable if h < r andNh is attracting. For n ≈ 0 (i.e. we are close to Nh),
n(t) ≈ exp [(h− r)t] .
This gives a recovery time as a function of h
TR(h) =1
r − hwith relative recovery time
TR(h)
TR(0)=
r
r − h.
In the case of maximal yield, i.e. h = r/2, this gives
TR(h = r/2)
TR(0)= 2 .
Stuart Townley Math Biol - Basics 20/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Does a dynamical analysis give us anything different?
Linearising (6) about the non-zero equilibrium Nh gives:
dn
dt= (f ′(Nh))n = (h− r)n .
Thus the non-zero equilibrium is linearly stable if h < r andNh is attracting. For n ≈ 0 (i.e. we are close to Nh),
n(t) ≈ exp [(h− r)t] .
This gives a recovery time as a function of h
TR(h) =1
r − hwith relative recovery time
TR(h)
TR(0)=
r
r − h.
In the case of maximal yield, i.e. h = r/2, this gives
TR(h = r/2)
TR(0)= 2 .
Stuart Townley Math Biol - Basics 20/ 54
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Does a dynamical analysis give us anything different?
Linearising (6) about the non-zero equilibrium Nh gives:
dn
dt= (f ′(Nh))n = (h− r)n .
Thus the non-zero equilibrium is linearly stable if h < r andNh is attracting. For n ≈ 0 (i.e. we are close to Nh),
n(t) ≈ exp [(h− r)t] .
This gives a recovery time as a function of h
TR(h) =1
r − hwith relative recovery time
TR(h)
TR(0)=
r
r − h.
In the case of maximal yield, i.e. h = r/2, this gives
TR(h = r/2)
TR(0)= 2 .
Stuart Townley Math Biol - Basics 20/ 54
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Does a dynamical analysis give us anything different?
Linearising (6) about the non-zero equilibrium Nh gives:
dn
dt= (f ′(Nh))n = (h− r)n .
Thus the non-zero equilibrium is linearly stable if h < r andNh is attracting. For n ≈ 0 (i.e. we are close to Nh),
n(t) ≈ exp [(h− r)t] .
This gives a recovery time as a function of h
TR(h) =1
r − hwith relative recovery time
TR(h)
TR(0)=
r
r − h.
In the case of maximal yield, i.e. h = r/2, this gives
TR(h = r/2)
TR(0)= 2 .
Stuart Townley Math Biol - Basics 20/ 54
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Does a dynamical analysis give us anything different?
Linearising (6) about the non-zero equilibrium Nh gives:
dn
dt= (f ′(Nh))n = (h− r)n .
Thus the non-zero equilibrium is linearly stable if h < r andNh is attracting. For n ≈ 0 (i.e. we are close to Nh),
n(t) ≈ exp [(h− r)t] .
This gives a recovery time as a function of h
TR(h) =1
r − hwith relative recovery time
TR(h)
TR(0)=
r
r − h.
In the case of maximal yield, i.e. h = r/2, this gives
TR(h = r/2)
TR(0)= 2 .
Stuart Townley Math Biol - Basics 20/ 54
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In this problem we are interested in yield Y = hN . At thenon-zero harvesting equilibrium N = Nh this means a yield
Y = hK(1− h
r) .
Solving for harvesting h in terms of yield Y gives:
h2 − rh+rY
K=⇒ h =
r
2
[1±
√1− 4Y
rK
]But Ymax = rK
4 . So
h =r
2
[1±
√1− Y
Ymax
]and relative recovery time:
TR(Y )
TR(0)=
2
1±√
1− YYmax
Stuart Townley Math Biol - Basics 21/ 54
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In this problem we are interested in yield Y = hN . At thenon-zero harvesting equilibrium N = Nh this means a yield
Y = hK(1− h
r) .
Solving for harvesting h in terms of yield Y gives:
h2 − rh+rY
K=⇒ h =
r
2
[1±
√1− 4Y
rK
]
But Ymax = rK4 . So
h =r
2
[1±
√1− Y
Ymax
]and relative recovery time:
TR(Y )
TR(0)=
2
1±√
1− YYmax
Stuart Townley Math Biol - Basics 21/ 54
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IntroductionContinuous time modelsDiscrete Population Models
In this problem we are interested in yield Y = hN . At thenon-zero harvesting equilibrium N = Nh this means a yield
Y = hK(1− h
r) .
Solving for harvesting h in terms of yield Y gives:
h2 − rh+rY
K=⇒ h =
r
2
[1±
√1− 4Y
rK
]But Ymax = rK
4 . So
h =r
2
[1±
√1− Y
Ymax
]and relative recovery time:
TR(Y )
TR(0)=
2
1±√
1− YYmax
Stuart Townley Math Biol - Basics 21/ 54
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0 0.5 1 1.50
1
2
3
4
5
6
7
8
9
10
0 0.5 1 1.50
1
2
3
4
5
6
7
8
9
10
Y/Ymax
Rel
ativ
e re
cove
ry ti
me
L−
L+
A
Stuart Townley Math Biol - Basics 22/ 54
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Starting with a small harvesting effort h ≈ 0, then Nh ≈ K,yield hNh is small, but the relative recovery time TR(Y )
TR(0) isclose to one and shortest
Then increasing h we follow the L+ branch in the figureabove. As h increases, Nh approaches K/2, the point A onthe graph, corresponding to maximum yield Y = Ymax, andrelative recovery time 2.
Increasing h beyond point A means we are now on the L−branch and the yield Y begins to decrease and recovery timeincreases.
So an optimal harvesting strategy is to choose h below r/2 sothat Nh is close to, but above, K/2. Getting too close toK/2 and with additional harvesting/model uncertainty risksswitching the dynamics onto the L− branch on which yieldthen diminishes.
Stuart Townley Math Biol - Basics 23/ 54
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IntroductionContinuous time modelsDiscrete Population Models
Starting with a small harvesting effort h ≈ 0, then Nh ≈ K,yield hNh is small, but the relative recovery time TR(Y )
TR(0) isclose to one and shortest
Then increasing h we follow the L+ branch in the figureabove. As h increases, Nh approaches K/2, the point A onthe graph, corresponding to maximum yield Y = Ymax, andrelative recovery time 2.
Increasing h beyond point A means we are now on the L−branch and the yield Y begins to decrease and recovery timeincreases.
So an optimal harvesting strategy is to choose h below r/2 sothat Nh is close to, but above, K/2. Getting too close toK/2 and with additional harvesting/model uncertainty risksswitching the dynamics onto the L− branch on which yieldthen diminishes.
Stuart Townley Math Biol - Basics 23/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Starting with a small harvesting effort h ≈ 0, then Nh ≈ K,yield hNh is small, but the relative recovery time TR(Y )
TR(0) isclose to one and shortest
Then increasing h we follow the L+ branch in the figureabove. As h increases, Nh approaches K/2, the point A onthe graph, corresponding to maximum yield Y = Ymax, andrelative recovery time 2.
Increasing h beyond point A means we are now on the L−branch and the yield Y begins to decrease and recovery timeincreases.
So an optimal harvesting strategy is to choose h below r/2 sothat Nh is close to, but above, K/2. Getting too close toK/2 and with additional harvesting/model uncertainty risksswitching the dynamics onto the L− branch on which yieldthen diminishes.
Stuart Townley Math Biol - Basics 23/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Starting with a small harvesting effort h ≈ 0, then Nh ≈ K,yield hNh is small, but the relative recovery time TR(Y )
TR(0) isclose to one and shortest
Then increasing h we follow the L+ branch in the figureabove. As h increases, Nh approaches K/2, the point A onthe graph, corresponding to maximum yield Y = Ymax, andrelative recovery time 2.
Increasing h beyond point A means we are now on the L−branch and the yield Y begins to decrease and recovery timeincreases.
So an optimal harvesting strategy is to choose h below r/2 sothat Nh is close to, but above, K/2. Getting too close toK/2 and with additional harvesting/model uncertainty risksswitching the dynamics onto the L− branch on which yieldthen diminishes.
Stuart Townley Math Biol - Basics 23/ 54
Overview of week aheadSingle Population Models
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IntroductionContinuous time modelsDiscrete Population Models
Many populations exhibit little overlap between successivegenerations (or more generally stage classes , see MBL2s).
In this case it is natural to model populations in discrete timesteps. Typically, the time step is a year, or a few days, or evenmilli-seconds.
Ignoring age- and/or stage-structured effects we arrive atmodels of the form:
Nt+1 = f(Nt) (8)
Different models for f will yield radically different populationdynamics.
Stuart Townley Math Biol - Basics 24/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Many populations exhibit little overlap between successivegenerations (or more generally stage classes , see MBL2s).
In this case it is natural to model populations in discrete timesteps. Typically, the time step is a year, or a few days, or evenmilli-seconds.
Ignoring age- and/or stage-structured effects we arrive atmodels of the form:
Nt+1 = f(Nt) (8)
Different models for f will yield radically different populationdynamics.
Stuart Townley Math Biol - Basics 24/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Many populations exhibit little overlap between successivegenerations (or more generally stage classes , see MBL2s).
In this case it is natural to model populations in discrete timesteps. Typically, the time step is a year, or a few days, or evenmilli-seconds.
Ignoring age- and/or stage-structured effects we arrive atmodels of the form:
Nt+1 = f(Nt) (8)
Different models for f will yield radically different populationdynamics.
Stuart Townley Math Biol - Basics 24/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
IntroductionContinuous time modelsDiscrete Population Models
Many populations exhibit little overlap between successivegenerations (or more generally stage classes , see MBL2s).
In this case it is natural to model populations in discrete timesteps. Typically, the time step is a year, or a few days, or evenmilli-seconds.
Ignoring age- and/or stage-structured effects we arrive atmodels of the form:
Nt+1 = f(Nt) (8)
Different models for f will yield radically different populationdynamics.
Stuart Townley Math Biol - Basics 24/ 54
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In general, crowding and self-regulation effects will lead to fhaving a maximum at some critical population density Nm
with f decreasing for n > Nm.
f Comments/Usefulness
rN(1− N
K
)Max. at N = K
2 , f(N) < 0 if N > K
rN exp[r(1− N
K
)]f(N) > 0, f decreases N large
Stuart Townley Math Biol - Basics 25/ 54
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In general, crowding and self-regulation effects will lead to fhaving a maximum at some critical population density Nm
with f decreasing for n > Nm.
f Comments/Usefulness
rN(1− N
K
)Max. at N = K
2 , f(N) < 0 if N > K
rN exp[r(1− N
K
)]f(N) > 0, f decreases N large
Stuart Townley Math Biol - Basics 25/ 54
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Cob-Webbing
The steady-states, N∗, of (8) satisfy f(N∗) = N∗
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
1.4
N
f(N)
f(N) = N
Stuart Townley Math Biol - Basics 26/ 54
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We can determine the solution Nt, t ≥ 0 using a graphicalCob-Webbing technique:
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
x=N
y=f(N)
N0 N1 N2
y=x
Stuart Townley Math Biol - Basics 27/ 54
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It is apparent that the slope of the graph N 7→ f(N) plays acrucial role in determining the qualitative dynamics of themodel.
The dynamics are distinguished by whether
f ′(N∗) ∈ (0, 1) (N∗ is stable),
f ′(N∗) ∈ (−1, 0) (N∗ is stable),
f ′(N∗) = −1 (N∗ is oscillatory), or
f ′(N∗) < −1 (N∗ is unstable).
This can be visualised using cob-web diagrams.
Stuart Townley Math Biol - Basics 28/ 54
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It is apparent that the slope of the graph N 7→ f(N) plays acrucial role in determining the qualitative dynamics of themodel.
The dynamics are distinguished by whether
f ′(N∗) ∈ (0, 1) (N∗ is stable),
f ′(N∗) ∈ (−1, 0) (N∗ is stable),
f ′(N∗) = −1 (N∗ is oscillatory), or
f ′(N∗) < −1 (N∗ is unstable).
This can be visualised using cob-web diagrams.
Stuart Townley Math Biol - Basics 28/ 54
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It is apparent that the slope of the graph N 7→ f(N) plays acrucial role in determining the qualitative dynamics of themodel.
The dynamics are distinguished by whether
f ′(N∗) ∈ (0, 1) (N∗ is stable),
f ′(N∗) ∈ (−1, 0) (N∗ is stable),
f ′(N∗) = −1 (N∗ is oscillatory), or
f ′(N∗) < −1 (N∗ is unstable).
This can be visualised using cob-web diagrams.
Stuart Townley Math Biol - Basics 28/ 54
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f ′(N∗) is a crucial quantity. We call it λ, the eigenvalue ofthe system
Nt+1 = f(Nt).
Ifλ ∈ (−1, 1) N∗ is an attracting eqilibrium,
|λ| > 1 N∗ is an unstable, repelling,
λ = −1 N∗ is a bifurcation value
λ = +1 N∗ is a bifurcation value.
Some extra reading: find out about bifurcations - saddle,pitchfork, ...
Stuart Townley Math Biol - Basics 29/ 54
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f ′(N∗) is a crucial quantity. We call it λ, the eigenvalue ofthe system
Nt+1 = f(Nt).
Ifλ ∈ (−1, 1) N∗ is an attracting eqilibrium,
|λ| > 1 N∗ is an unstable, repelling,
λ = −1 N∗ is a bifurcation value
λ = +1 N∗ is a bifurcation value.
Some extra reading: find out about bifurcations - saddle,pitchfork, ...
Stuart Townley Math Biol - Basics 29/ 54
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f ′(N∗) is a crucial quantity. We call it λ, the eigenvalue ofthe system
Nt+1 = f(Nt).
Ifλ ∈ (−1, 1) N∗ is an attracting eqilibrium,
|λ| > 1 N∗ is an unstable, repelling,
λ = −1 N∗ is a bifurcation value
λ = +1 N∗ is a bifurcation value.
Some extra reading: find out about bifurcations - saddle,pitchfork, ...
Stuart Townley Math Biol - Basics 29/ 54
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IntroductionContinuous time modelsDiscrete Population Models
Chaos in the logistics map.
Setting ut = Nt/K, then the discrete logistics map:
Nt+1 = rNt
(1− Nt
K
)becomes:
ut+1 = rut(1− ut) .
With f(u) = ru(1− u), f ′(u) = r(1− 2u).
We have steady-states u∗ given by
u∗ = ru∗(1− u∗), i.e. u∗ = 0, u∗ =r − 1
r
At u∗ = 0, λ = f ′(0) = r. At u∗ = r−1r , with necessarily
r > 1, λ = f ′( r−1r ) = 2− r < 1.
Stuart Townley Math Biol - Basics 30/ 54
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Chaos in the logistics map.
Setting ut = Nt/K, then the discrete logistics map:
Nt+1 = rNt
(1− Nt
K
)becomes:
ut+1 = rut(1− ut) .
With f(u) = ru(1− u), f ′(u) = r(1− 2u).
We have steady-states u∗ given by
u∗ = ru∗(1− u∗), i.e. u∗ = 0, u∗ =r − 1
r
At u∗ = 0, λ = f ′(0) = r. At u∗ = r−1r , with necessarily
r > 1, λ = f ′( r−1r ) = 2− r < 1.
Stuart Townley Math Biol - Basics 30/ 54
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Chaos in the logistics map.
Setting ut = Nt/K, then the discrete logistics map:
Nt+1 = rNt
(1− Nt
K
)becomes:
ut+1 = rut(1− ut) .
With f(u) = ru(1− u), f ′(u) = r(1− 2u).
We have steady-states u∗ given by
u∗ = ru∗(1− u∗), i.e. u∗ = 0, u∗ =r − 1
r
At u∗ = 0, λ = f ′(0) = r. At u∗ = r−1r , with necessarily
r > 1, λ = f ′( r−1r ) = 2− r < 1.
Stuart Townley Math Biol - Basics 30/ 54
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Chaos in the logistics map.
Setting ut = Nt/K, then the discrete logistics map:
Nt+1 = rNt
(1− Nt
K
)becomes:
ut+1 = rut(1− ut) .
With f(u) = ru(1− u), f ′(u) = r(1− 2u).
We have steady-states u∗ given by
u∗ = ru∗(1− u∗), i.e. u∗ = 0, u∗ =r − 1
r
At u∗ = 0, λ = f ′(0) = r. At u∗ = r−1r , with necessarily
r > 1, λ = f ′( r−1r ) = 2− r < 1.
Stuart Townley Math Biol - Basics 30/ 54
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We can analyse this model using r as a bifurcation parameter,starting with small r ≈ 0 and gradually increasing it.
0 < r < 1: u∗ = 0 is a unique equilibrium with λ = r < 1. Sou∗ in this case is STABLE.
r = 1. First bifurcation. u∗ loses its stability and we gain anew non-zero equilbrium u∗ = r−1
r .
1 < r < 3. u∗ = 0 is unstable. u∗ = r−1r , with λ = 2− r, is
stable.
r = 3. Here there is a pitchfork bifurcation. u∗ = r−1r loses
stability. What happens next?
Stuart Townley Math Biol - Basics 31/ 54
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We can analyse this model using r as a bifurcation parameter,starting with small r ≈ 0 and gradually increasing it.
0 < r < 1: u∗ = 0 is a unique equilibrium with λ = r < 1. Sou∗ in this case is STABLE.
r = 1. First bifurcation. u∗ loses its stability and we gain anew non-zero equilbrium u∗ = r−1
r .
1 < r < 3. u∗ = 0 is unstable. u∗ = r−1r , with λ = 2− r, is
stable.
r = 3. Here there is a pitchfork bifurcation. u∗ = r−1r loses
stability. What happens next?
Stuart Townley Math Biol - Basics 31/ 54
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IntroductionContinuous time modelsDiscrete Population Models
We can analyse this model using r as a bifurcation parameter,starting with small r ≈ 0 and gradually increasing it.
0 < r < 1: u∗ = 0 is a unique equilibrium with λ = r < 1. Sou∗ in this case is STABLE.
r = 1. First bifurcation. u∗ loses its stability and we gain anew non-zero equilbrium u∗ = r−1
r .
1 < r < 3. u∗ = 0 is unstable. u∗ = r−1r , with λ = 2− r, is
stable.
r = 3. Here there is a pitchfork bifurcation. u∗ = r−1r loses
stability. What happens next?
Stuart Townley Math Biol - Basics 31/ 54
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IntroductionContinuous time modelsDiscrete Population Models
We can analyse this model using r as a bifurcation parameter,starting with small r ≈ 0 and gradually increasing it.
0 < r < 1: u∗ = 0 is a unique equilibrium with λ = r < 1. Sou∗ in this case is STABLE.
r = 1. First bifurcation. u∗ loses its stability and we gain anew non-zero equilbrium u∗ = r−1
r .
1 < r < 3. u∗ = 0 is unstable. u∗ = r−1r , with λ = 2− r, is
stable.
r = 3. Here there is a pitchfork bifurcation. u∗ = r−1r loses
stability. What happens next?
Stuart Townley Math Biol - Basics 31/ 54
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IntroductionContinuous time modelsDiscrete Population Models
We can analyse this model using r as a bifurcation parameter,starting with small r ≈ 0 and gradually increasing it.
0 < r < 1: u∗ = 0 is a unique equilibrium with λ = r < 1. Sou∗ in this case is STABLE.
r = 1. First bifurcation. u∗ loses its stability and we gain anew non-zero equilbrium u∗ = r−1
r .
1 < r < 3. u∗ = 0 is unstable. u∗ = r−1r , with λ = 2− r, is
stable.
r = 3. Here there is a pitchfork bifurcation. u∗ = r−1r loses
stability. What happens next?
Stuart Townley Math Biol - Basics 31/ 54
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Let’s seek period 2 solutions - i.e. solutions with
ut+2 = ut, but with ut+1 not necessarily equal to ut
So we need to look for
ut+2 = f(ut+1) = f(f(ut))
= f(rut[1− ut])
= r(rut[1− ut])(1− rut[1− ut])
This iteration has equilibria ut+2 = ut = u∗ given by
u∗ = r2u∗[1− u∗](1− ru∗[1− u∗]) (9)
Stuart Townley Math Biol - Basics 32/ 54
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Let’s seek period 2 solutions - i.e. solutions with
ut+2 = ut, but with ut+1 not necessarily equal to ut
So we need to look for
ut+2 = f(ut+1) = f(f(ut))
= f(rut[1− ut])
= r(rut[1− ut])(1− rut[1− ut])
This iteration has equilibria ut+2 = ut = u∗ given by
u∗ = r2u∗[1− u∗](1− ru∗[1− u∗]) (9)
Stuart Townley Math Biol - Basics 32/ 54
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Let’s seek period 2 solutions - i.e. solutions with
ut+2 = ut, but with ut+1 not necessarily equal to ut
So we need to look for
ut+2 = f(ut+1) = f(f(ut))
= f(rut[1− ut])
= r(rut[1− ut])(1− rut[1− ut])
This iteration has equilibria ut+2 = ut = u∗ given by
u∗ = r2u∗[1− u∗](1− ru∗[1− u∗]) (9)
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The quartic equation
u∗ = r2u∗[1− u∗](1− ru∗[1− u∗])contains the equilibrium solutions u∗ = 0, u∗ = r−1
r .
In fact (9) factorises to give
u∗[ru∗ − (r − 1)][r2(u∗)2 − r(r + 1)u∗ + (r + 1)] = 0
In addition to the equilibria u∗ = 0, and if r > 1, u∗ = r−1r ,
this yields period 2 solutions
u∗ =(r + 1)±
√(r + 1)(r − 3)
2r, valid for r > 3 .
As r passes through r = 3, we develop TWO period 2solutions. Are these stable or not? We need to linearise theperiod two dynamics:
ut+2 = f(f(ut)) = g(ut), with g(u) = r2u[1−u](1−ru[1−u]) .
Stuart Townley Math Biol - Basics 33/ 54
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The quartic equation
u∗ = r2u∗[1− u∗](1− ru∗[1− u∗])contains the equilibrium solutions u∗ = 0, u∗ = r−1
r .In fact (9) factorises to give
u∗[ru∗ − (r − 1)][r2(u∗)2 − r(r + 1)u∗ + (r + 1)] = 0
In addition to the equilibria u∗ = 0, and if r > 1, u∗ = r−1r ,
this yields period 2 solutions
u∗ =(r + 1)±
√(r + 1)(r − 3)
2r, valid for r > 3 .
As r passes through r = 3, we develop TWO period 2solutions. Are these stable or not? We need to linearise theperiod two dynamics:
ut+2 = f(f(ut)) = g(ut), with g(u) = r2u[1−u](1−ru[1−u]) .
Stuart Townley Math Biol - Basics 33/ 54
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The quartic equation
u∗ = r2u∗[1− u∗](1− ru∗[1− u∗])contains the equilibrium solutions u∗ = 0, u∗ = r−1
r .In fact (9) factorises to give
u∗[ru∗ − (r − 1)][r2(u∗)2 − r(r + 1)u∗ + (r + 1)] = 0
In addition to the equilibria u∗ = 0, and if r > 1, u∗ = r−1r ,
this yields period 2 solutions
u∗ =(r + 1)±
√(r + 1)(r − 3)
2r, valid for r > 3 .
As r passes through r = 3, we develop TWO period 2solutions. Are these stable or not? We need to linearise theperiod two dynamics:
ut+2 = f(f(ut)) = g(ut), with g(u) = r2u[1−u](1−ru[1−u]) .
Stuart Townley Math Biol - Basics 33/ 54
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The quartic equation
u∗ = r2u∗[1− u∗](1− ru∗[1− u∗])contains the equilibrium solutions u∗ = 0, u∗ = r−1
r .In fact (9) factorises to give
u∗[ru∗ − (r − 1)][r2(u∗)2 − r(r + 1)u∗ + (r + 1)] = 0
In addition to the equilibria u∗ = 0, and if r > 1, u∗ = r−1r ,
this yields period 2 solutions
u∗ =(r + 1)±
√(r + 1)(r − 3)
2r, valid for r > 3 .
As r passes through r = 3, we develop TWO period 2solutions. Are these stable or not? We need to linearise theperiod two dynamics:
ut+2 = f(f(ut)) = g(ut), with g(u) = r2u[1−u](1−ru[1−u]) .
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The eigenvalues for these period 2 solutions are given by
λ = g′(u)|u=u∗
One can show (quite easily) that λ ∈ (−1,+1), for3 < r < 1 +
√6 = 3.4495.
For r > 3.4495, we seek stable period 4 solutions, which as rincreases further still break down into period 8, 16, ...solutions.
This period doubling process continues as r increases toproduce stable solutions of higher and higher periods. There isa critical value r ≈ 3.89 beyond which all period 2n solutions,for all n, are unstable. This is the chaotic regime.
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The eigenvalues for these period 2 solutions are given by
λ = g′(u)|u=u∗
One can show (quite easily) that λ ∈ (−1,+1), for3 < r < 1 +
√6 = 3.4495.
For r > 3.4495, we seek stable period 4 solutions, which as rincreases further still break down into period 8, 16, ...solutions.
This period doubling process continues as r increases toproduce stable solutions of higher and higher periods. There isa critical value r ≈ 3.89 beyond which all period 2n solutions,for all n, are unstable. This is the chaotic regime.
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The eigenvalues for these period 2 solutions are given by
λ = g′(u)|u=u∗
One can show (quite easily) that λ ∈ (−1,+1), for3 < r < 1 +
√6 = 3.4495.
For r > 3.4495, we seek stable period 4 solutions, which as rincreases further still break down into period 8, 16, ...solutions.
This period doubling process continues as r increases toproduce stable solutions of higher and higher periods. There isa critical value r ≈ 3.89 beyond which all period 2n solutions,for all n, are unstable. This is the chaotic regime.
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IntroductionContinuous time modelsDiscrete Population Models
The eigenvalues for these period 2 solutions are given by
λ = g′(u)|u=u∗
One can show (quite easily) that λ ∈ (−1,+1), for3 < r < 1 +
√6 = 3.4495.
For r > 3.4495, we seek stable period 4 solutions, which as rincreases further still break down into period 8, 16, ...solutions.
This period doubling process continues as r increases toproduce stable solutions of higher and higher periods. There isa critical value r ≈ 3.89 beyond which all period 2n solutions,for all n, are unstable. This is the chaotic regime.
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IntroductionContinuous time modelsDiscrete Population Models
Stuart Townley Math Biol - Basics 35/ 54
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1 Pedator–prey models
2 Competition and mutualism
Revision: Think phase plane analysis
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Many species interact. For two species these interactions takeone of three distinct forms:
If the growth rate of one of the species is decreased by the(presence of the) other we call this a PREDATOR-PREYsystem.
If the growth rates of each species are decreased by eachother we call this a COMPETITIVE system.
If the growth rates of each species are enhanced by each otherwe call this a MUTUALISTIC system.
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IntroductionPredator–prey modelsCompetition and Cooperation
Many species interact. For two species these interactions takeone of three distinct forms:
If the growth rate of one of the species is decreased by the(presence of the) other we call this a PREDATOR-PREYsystem.
If the growth rates of each species are decreased by eachother we call this a COMPETITIVE system.
If the growth rates of each species are enhanced by each otherwe call this a MUTUALISTIC system.
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IntroductionPredator–prey modelsCompetition and Cooperation
Many species interact. For two species these interactions takeone of three distinct forms:
If the growth rate of one of the species is decreased by the(presence of the) other we call this a PREDATOR-PREYsystem.
If the growth rates of each species are decreased by eachother we call this a COMPETITIVE system.
If the growth rates of each species are enhanced by each otherwe call this a MUTUALISTIC system.
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IntroductionPredator–prey modelsCompetition and Cooperation
Many species interact. For two species these interactions takeone of three distinct forms:
If the growth rate of one of the species is decreased by the(presence of the) other we call this a PREDATOR-PREYsystem.
If the growth rates of each species are decreased by eachother we call this a COMPETITIVE system.
If the growth rates of each species are enhanced by each otherwe call this a MUTUALISTIC system.
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The Lotka-Volterra predator–prey system
In 1926 Lotka/Volterra proposed a simple model to explainthe oscillatory levels of certain fish catches in the Adriatic.If N(t) and P (t) are the populations of the prey and thepredator then
dN
dt= N(a− bP ) dP
dt= P (cN − d) (10)
a, b, c and d are constant parameters and
In the absence of predators, dNdt ∝ N
Predation reduces the prey population growth rate by anamount proportional to both N and P .
The presence of prey increases the population growth rate ofpredators by an amount proportional to N and P .
In the absence of prey, dPdt ∝ P .
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The Lotka-Volterra predator–prey system
In 1926 Lotka/Volterra proposed a simple model to explainthe oscillatory levels of certain fish catches in the Adriatic.If N(t) and P (t) are the populations of the prey and thepredator then
dN
dt= N(a− bP ) dP
dt= P (cN − d) (10)
a, b, c and d are constant parameters and
In the absence of predators, dNdt ∝ N
Predation reduces the prey population growth rate by anamount proportional to both N and P .
The presence of prey increases the population growth rate ofpredators by an amount proportional to N and P .
In the absence of prey, dPdt ∝ P .
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IntroductionPredator–prey modelsCompetition and Cooperation
The Lotka-Volterra predator–prey system
In 1926 Lotka/Volterra proposed a simple model to explainthe oscillatory levels of certain fish catches in the Adriatic.If N(t) and P (t) are the populations of the prey and thepredator then
dN
dt= N(a− bP ) dP
dt= P (cN − d) (10)
a, b, c and d are constant parameters and
In the absence of predators, dNdt ∝ N
Predation reduces the prey population growth rate by anamount proportional to both N and P .
The presence of prey increases the population growth rate ofpredators by an amount proportional to N and P .
In the absence of prey, dPdt ∝ P .
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IntroductionPredator–prey modelsCompetition and Cooperation
The Lotka-Volterra predator–prey system
In 1926 Lotka/Volterra proposed a simple model to explainthe oscillatory levels of certain fish catches in the Adriatic.If N(t) and P (t) are the populations of the prey and thepredator then
dN
dt= N(a− bP ) dP
dt= P (cN − d) (10)
a, b, c and d are constant parameters and
In the absence of predators, dNdt ∝ N
Predation reduces the prey population growth rate by anamount proportional to both N and P .
The presence of prey increases the population growth rate ofpredators by an amount proportional to N and P .
In the absence of prey, dPdt ∝ P .
Stuart Townley Math Biol - Basics 38/ 54
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IntroductionPredator–prey modelsCompetition and Cooperation
The Lotka-Volterra predator–prey system
In 1926 Lotka/Volterra proposed a simple model to explainthe oscillatory levels of certain fish catches in the Adriatic.If N(t) and P (t) are the populations of the prey and thepredator then
dN
dt= N(a− bP ) dP
dt= P (cN − d) (10)
a, b, c and d are constant parameters and
In the absence of predators, dNdt ∝ N
Predation reduces the prey population growth rate by anamount proportional to both N and P .
The presence of prey increases the population growth rate ofpredators by an amount proportional to N and P .
In the absence of prey, dPdt ∝ P .
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We non-dimensionalize to reduce parameters by writing
u(τ) =c
dN(t) ν(τ) =
b
aP (t) τ = at α =
d
a.
This gives
du
dτ= u(1− v) dv
dτ= αv(u− 1) (11)
The non-zero equilibrium is u = v = 1
In a phase plane dvdu = αv(u−1)
u(1−v) . Separating variable implies
(1− v)v
dv = α(u− 1)
udu
Then
αu+ v − ln(uαv) = H (H is a constant)
For a given H > 1+α these are closed trajectories, see figure.
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We non-dimensionalize to reduce parameters by writing
u(τ) =c
dN(t) ν(τ) =
b
aP (t) τ = at α =
d
a.
This gives
du
dτ= u(1− v) dv
dτ= αv(u− 1) (11)
The non-zero equilibrium is u = v = 1
In a phase plane dvdu = αv(u−1)
u(1−v) . Separating variable implies
(1− v)v
dv = α(u− 1)
udu
Then
αu+ v − ln(uαv) = H (H is a constant)
For a given H > 1+α these are closed trajectories, see figure.
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We non-dimensionalize to reduce parameters by writing
u(τ) =c
dN(t) ν(τ) =
b
aP (t) τ = at α =
d
a.
This gives
du
dτ= u(1− v) dv
dτ= αv(u− 1) (11)
The non-zero equilibrium is u = v = 1
In a phase plane dvdu = αv(u−1)
u(1−v) . Separating variable implies
(1− v)v
dv = α(u− 1)
udu
Then
αu+ v − ln(uαv) = H (H is a constant)
For a given H > 1+α these are closed trajectories, see figure.
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We non-dimensionalize to reduce parameters by writing
u(τ) =c
dN(t) ν(τ) =
b
aP (t) τ = at α =
d
a.
This gives
du
dτ= u(1− v) dv
dτ= αv(u− 1) (11)
The non-zero equilibrium is u = v = 1
In a phase plane dvdu = αv(u−1)
u(1−v) . Separating variable implies
(1− v)v
dv = α(u− 1)
udu
Then
αu+ v − ln(uαv) = H (H is a constant)
For a given H > 1+α these are closed trajectories, see figure.
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u
v
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
4
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Competition Models
Here two or more species compete for the same limited foodsource or in some way inhibit each others growth.
For example:
dN1dt = r1N1
[1− N1
K1− b12N2
K1
]dN2dt = r2N2
[1− N2
K2− b21N1
K2
] (12)
Here both species grow logistically in the absence of the otherthe positive parameters b12 and b21 indicate the inhibitioneffect
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Competition Models
Here two or more species compete for the same limited foodsource or in some way inhibit each others growth.
For example:
dN1dt = r1N1
[1− N1
K1− b12N2
K1
]dN2dt = r2N2
[1− N2
K2− b21N1
K2
] (12)
Here both species grow logistically in the absence of the otherthe positive parameters b12 and b21 indicate the inhibitioneffect
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Competition Models
Here two or more species compete for the same limited foodsource or in some way inhibit each others growth.
For example:
dN1dt = r1N1
[1− N1
K1− b12N2
K1
]dN2dt = r2N2
[1− N2
K2− b21N1
K2
] (12)
Here both species grow logistically in the absence of the otherthe positive parameters b12 and b21 indicate the inhibitioneffect
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We nondimensionalise by writing
ui =Ni
Ki, τ = r1t, ρ =
r2r1
α12 =b12K2
K1α21 =
b21K1
K2
du1dτ = u1[1− u1 − α12u2] = f(u1, u2)
du2dτ = ρu2[1− u2 − α21u1] = g(u1, u2)
(13)
The steady states are (exercise)
(u∗1, u∗2) = (0, 0), (1, 0), (0, 1),
(1− α12
1− α12α21,
1− α21
1− α12α21
)The null clines (lines on which f = 0 or g = 0) can bearranged in 4 ways depending on the relative sizes of α12 andα21
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We nondimensionalise by writing
ui =Ni
Ki, τ = r1t, ρ =
r2r1
α12 =b12K2
K1α21 =
b21K1
K2
du1dτ = u1[1− u1 − α12u2] = f(u1, u2)
du2dτ = ρu2[1− u2 − α21u1] = g(u1, u2)
(13)
The steady states are (exercise)
(u∗1, u∗2) = (0, 0), (1, 0), (0, 1),
(1− α12
1− α12α21,
1− α21
1− α12α21
)The null clines (lines on which f = 0 or g = 0) can bearranged in 4 ways depending on the relative sizes of α12 andα21
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We nondimensionalise by writing
ui =Ni
Ki, τ = r1t, ρ =
r2r1
α12 =b12K2
K1α21 =
b21K1
K2
du1dτ = u1[1− u1 − α12u2] = f(u1, u2)
du2dτ = ρu2[1− u2 − α21u1] = g(u1, u2)
(13)
The steady states are (exercise)
(u∗1, u∗2) = (0, 0), (1, 0), (0, 1),
(1− α12
1− α12α21,
1− α21
1− α12α21
)The null clines (lines on which f = 0 or g = 0) can bearranged in 4 ways depending on the relative sizes of α12 andα21
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u1
u2
u2
u2
u2
u1
u1 u1
s
u
u
u
s
su
u
uuu
u
s
s
s - stable equilibrium, u - unstable equilibrium.(a) Top Left α12, α21 > 1 (b) Top Right α12, α21 < 1 (c) BottomLeft α12 < 1, α21 > 1 (d) Bottom Right α12 > 1, α21 < 1
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In cases c) and d) we have no co-existent steady states, (i.e.when an equilibrium (u∗1, u
∗2) has u∗1 and u∗2 both non-zero)
If dudt = f(u) then to look at the stability of the various states
we need to look at the community matrix
A =
(∂f∂u1
∂f∂u2
∂g∂u1
∂g∂u2
)=
(1− 2u1 − α12u2 −α12u1−ρα21u2 ρ(1− 2u2 − α21u1)
)
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In cases c) and d) we have no co-existent steady states, (i.e.when an equilibrium (u∗1, u
∗2) has u∗1 and u∗2 both non-zero)
If dudt = f(u) then to look at the stability of the various states
we need to look at the community matrix
A =
(∂f∂u1
∂f∂u2
∂g∂u1
∂g∂u2
)=
(1− 2u1 − α12u2 −α12u1−ρα21u2 ρ(1− 2u2 − α21u1)
)
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(0, 0). Now A =
(1 00 ρ
)=⇒ λ = 1, ρ. (0, 0) is an
UNSTABLE NODE
(1, 0). Now A =
(−1 −α12
0 ρ(1− α21)
)so λ = −1, ρ(1− α21)
(1, 0) is an STABLE NODE if α21 > 1(1, 0) is an UNSTABLE SADDLE if α21 < 1
(0, 1). Now A =
(1− α12 0−ρα −ρ
)=⇒ λ = −ρ, (1− α12)
(0, 1) is a STABLE NODE if α12 > 1(0, 1) is an UNSTABLE SADDLE if α12 < 1
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(0, 0). Now A =
(1 00 ρ
)=⇒ λ = 1, ρ. (0, 0) is an
UNSTABLE NODE
(1, 0). Now A =
(−1 −α12
0 ρ(1− α21)
)so λ = −1, ρ(1− α21)
(1, 0) is an STABLE NODE if α21 > 1(1, 0) is an UNSTABLE SADDLE if α21 < 1
(0, 1). Now A =
(1− α12 0−ρα −ρ
)=⇒ λ = −ρ, (1− α12)
(0, 1) is a STABLE NODE if α12 > 1(0, 1) is an UNSTABLE SADDLE if α12 < 1
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(0, 0). Now A =
(1 00 ρ
)=⇒ λ = 1, ρ. (0, 0) is an
UNSTABLE NODE
(1, 0). Now A =
(−1 −α12
0 ρ(1− α21)
)so λ = −1, ρ(1− α21)
(1, 0) is an STABLE NODE if α21 > 1(1, 0) is an UNSTABLE SADDLE if α21 < 1
(0, 1). Now A =
(1− α12 0−ρα −ρ
)=⇒ λ = −ρ, (1− α12)
(0, 1) is a STABLE NODE if α12 > 1(0, 1) is an UNSTABLE SADDLE if α12 < 1
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For the equilibrium:(
1−α121−α12α21
, 1−α211−α12α21
). Needs
α12, α21 < 1 or α21, α12 > 1
A =
(−u∗1 −α12u
∗1
−ρα21u∗2 −ρu∗2
)(14)
Trace(A) = −u∗1 − ρu∗2 = λ1 + λ2 < 0Det(A) = ρu∗1u
∗2 − ρα12α21u)1
∗u∗2= ρu∗1u
∗2(1− α12α21)
= λ1λ2
if α12α21 < 1 Re(λ) < 0 =⇒ STABLEif α12α21 > 1 λ ∈ R with λ1 < 0 < λ2 =⇒ UNSTABLESADDLE
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For the equilibrium:(
1−α121−α12α21
, 1−α211−α12α21
). Needs
α12, α21 < 1 or α21, α12 > 1
A =
(−u∗1 −α12u
∗1
−ρα21u∗2 −ρu∗2
)(14)
Trace(A) = −u∗1 − ρu∗2 = λ1 + λ2 < 0Det(A) = ρu∗1u
∗2 − ρα12α21u)1
∗u∗2= ρu∗1u
∗2(1− α12α21)
= λ1λ2
if α12α21 < 1 Re(λ) < 0 =⇒ STABLEif α12α21 > 1 λ ∈ R with λ1 < 0 < λ2 =⇒ UNSTABLESADDLE
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For the equilibrium:(
1−α121−α12α21
, 1−α211−α12α21
). Needs
α12, α21 < 1 or α21, α12 > 1
A =
(−u∗1 −α12u
∗1
−ρα21u∗2 −ρu∗2
)(14)
Trace(A) = −u∗1 − ρu∗2 = λ1 + λ2 < 0Det(A) = ρu∗1u
∗2 − ρα12α21u)1
∗u∗2= ρu∗1u
∗2(1− α12α21)
= λ1λ2
if α12α21 < 1 Re(λ) < 0 =⇒ STABLEif α12α21 > 1 λ ∈ R with λ1 < 0 < λ2 =⇒ UNSTABLESADDLE
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For the equilibrium:(
1−α121−α12α21
, 1−α211−α12α21
). Needs
α12, α21 < 1 or α21, α12 > 1
A =
(−u∗1 −α12u
∗1
−ρα21u∗2 −ρu∗2
)(14)
Trace(A) = −u∗1 − ρu∗2 = λ1 + λ2 < 0Det(A) = ρu∗1u
∗2 − ρα12α21u)1
∗u∗2= ρu∗1u
∗2(1− α12α21)
= λ1λ2
if α12α21 < 1 Re(λ) < 0 =⇒ STABLEif α12α21 > 1 λ ∈ R with λ1 < 0 < λ2 =⇒ UNSTABLESADDLE
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The fact that the stable steady state is either (0, 1) or (1, 0)illustrates the principle of competitive exclusion
If two species compete according to (12), then one of thembecomes extinct.
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The fact that the stable steady state is either (0, 1) or (1, 0)illustrates the principle of competitive exclusion
If two species compete according to (12), then one of thembecomes extinct.
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Mutualism
There are many real-life situations where the interaction oftwo or more species is to the advantage of all.
A typical model is
dN1dt = r1N1
(1− N1
K1+ b12
N2K1
)dN2dt = r2N2
(1− N2
K2+ b21
N1K2
)The +N1N2 in both RHS models that both species increasein the presence of the other.
Let ui =NiKi
τ = r1t ρ = r2r1
α12 =b12K2K1
α21 =b21K1K2
thendu1dτ = u1(1− u1 + α12u2)du2dτ = ρu2(1− u2 + α21u1)
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Mutualism
There are many real-life situations where the interaction oftwo or more species is to the advantage of all.
A typical model is
dN1dt = r1N1
(1− N1
K1+ b12
N2K1
)dN2dt = r2N2
(1− N2
K2+ b21
N1K2
)The +N1N2 in both RHS models that both species increasein the presence of the other.
Let ui =NiKi
τ = r1t ρ = r2r1
α12 =b12K2K1
α21 =b21K1K2
thendu1dτ = u1(1− u1 + α12u2)du2dτ = ρu2(1− u2 + α21u1)
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IntroductionPredator–prey modelsCompetition and Cooperation
Mutualism
There are many real-life situations where the interaction oftwo or more species is to the advantage of all.
A typical model is
dN1dt = r1N1
(1− N1
K1+ b12
N2K1
)dN2dt = r2N2
(1− N2
K2+ b21
N1K2
)The +N1N2 in both RHS models that both species increasein the presence of the other.
Let ui =NiKi
τ = r1t ρ = r2r1
α12 =b12K2K1
α21 =b21K1K2
thendu1dτ = u1(1− u1 + α12u2)du2dτ = ρu2(1− u2 + α21u1)
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The steady states are
(0, 0), (1, 0), (0, 1) and
(1 + α12
1− α12α21,
1 + α21
1− α12α21
)(if α12α21 < 1)
Community matrix
A =
[1− 2u1 + α12u2 α12u1
ρα21u2 ρ(1− 2u2 + α21u1)
]
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The steady states are
(0, 0), (1, 0), (0, 1) and
(1 + α12
1− α12α21,
1 + α21
1− α12α21
)(if α12α21 < 1)
Community matrix
A =
[1− 2u1 + α12u2 α12u1
ρα21u2 ρ(1− 2u2 + α21u1)
]
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(0, 0) A =
(1 00 ρ
)λ = 1, ρ. (0, 0) is an UNSTABLE
NODE
(1, 0) A =
(−1 α12
0 ρ(1 + α21)
)λ = −1, ρ(1 + α21)
(1, 0) is an UNSTABLE SADDLE
(0, 1) A =
(1 + α12 0ρα21 −ρ
)λ = −ρ, 1 + α12 (0, 1) is an
UNSTABLE SADDLE
(u∗1, u∗2) =
(1+α12
1−α12α21, 1+α211−α12α21
)=⇒ A =
(−u∗1 α12u
∗1
ρα12u∗2 −ρu∗2
)traceA < 0, det(A) = ρ(1− α12α21)u
∗1u∗2 > 0
so when it exists it is STABLE
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IntroductionPredator–prey modelsCompetition and Cooperation
(0, 0) A =
(1 00 ρ
)λ = 1, ρ. (0, 0) is an UNSTABLE
NODE
(1, 0) A =
(−1 α12
0 ρ(1 + α21)
)λ = −1, ρ(1 + α21)
(1, 0) is an UNSTABLE SADDLE
(0, 1) A =
(1 + α12 0ρα21 −ρ
)λ = −ρ, 1 + α12 (0, 1) is an
UNSTABLE SADDLE
(u∗1, u∗2) =
(1+α12
1−α12α21, 1+α211−α12α21
)=⇒ A =
(−u∗1 α12u
∗1
ρα12u∗2 −ρu∗2
)traceA < 0, det(A) = ρ(1− α12α21)u
∗1u∗2 > 0
so when it exists it is STABLE
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Overview of week aheadSingle Population Models
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IntroductionPredator–prey modelsCompetition and Cooperation
(0, 0) A =
(1 00 ρ
)λ = 1, ρ. (0, 0) is an UNSTABLE
NODE
(1, 0) A =
(−1 α12
0 ρ(1 + α21)
)λ = −1, ρ(1 + α21)
(1, 0) is an UNSTABLE SADDLE
(0, 1) A =
(1 + α12 0ρα21 −ρ
)λ = −ρ, 1 + α12 (0, 1) is an
UNSTABLE SADDLE
(u∗1, u∗2) =
(1+α12
1−α12α21, 1+α211−α12α21
)=⇒ A =
(−u∗1 α12u
∗1
ρα12u∗2 −ρu∗2
)traceA < 0, det(A) = ρ(1− α12α21)u
∗1u∗2 > 0
so when it exists it is STABLE
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Overview of week aheadSingle Population Models
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IntroductionPredator–prey modelsCompetition and Cooperation
(0, 0) A =
(1 00 ρ
)λ = 1, ρ. (0, 0) is an UNSTABLE
NODE
(1, 0) A =
(−1 α12
0 ρ(1 + α21)
)λ = −1, ρ(1 + α21)
(1, 0) is an UNSTABLE SADDLE
(0, 1) A =
(1 + α12 0ρα21 −ρ
)λ = −ρ, 1 + α12 (0, 1) is an
UNSTABLE SADDLE
(u∗1, u∗2) =
(1+α12
1−α12α21, 1+α211−α12α21
)=⇒ A =
(−u∗1 α12u
∗1
ρα12u∗2 −ρu∗2
)traceA < 0, det(A) = ρ(1− α12α21)u
∗1u∗2 > 0
so when it exists it is STABLE
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u1
u2
u1
u2
u
u u0
s
u
u
u0
1
1
1
1
u - unstable equilibrium s - stable equilibrium
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For the mutualistic model
du1dτ = u1(1− u1 + α12u2)
du2dτ = ρu2(1− u2 + α21u1)
if α12α21 > 1, all equilibrium points unstable and unboundedgrowth in each species.
if α12α21 < 1 new steady state is stable and system evolves toa co-existent state.
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Overview of week aheadSingle Population Models
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IntroductionPredator–prey modelsCompetition and Cooperation
For the mutualistic model
du1dτ = u1(1− u1 + α12u2)
du2dτ = ρu2(1− u2 + α21u1)
if α12α21 > 1, all equilibrium points unstable and unboundedgrowth in each species.
if α12α21 < 1 new steady state is stable and system evolves toa co-existent state.
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Overview of week aheadSingle Population Models
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IntroductionPredator–prey modelsCompetition and Cooperation
For the mutualistic model
du1dτ = u1(1− u1 + α12u2)
du2dτ = ρu2(1− u2 + α21u1)
if α12α21 > 1, all equilibrium points unstable and unboundedgrowth in each species.
if α12α21 < 1 new steady state is stable and system evolves toa co-existent state.
Stuart Townley Math Biol - Basics 52/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
single (scalar) and two species models
continuous and discrete time
nonlinearity, i.e density dependence
equilibria and local stability
harvesting, i.e. control
Stuart Townley Math Biol - Basics 53/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
single (scalar) and two species models
continuous and discrete time
nonlinearity, i.e density dependence
equilibria and local stability
harvesting, i.e. control
Stuart Townley Math Biol - Basics 53/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
single (scalar) and two species models
continuous and discrete time
nonlinearity, i.e density dependence
equilibria and local stability
harvesting, i.e. control
Stuart Townley Math Biol - Basics 53/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
single (scalar) and two species models
continuous and discrete time
nonlinearity, i.e density dependence
equilibria and local stability
harvesting, i.e. control
Stuart Townley Math Biol - Basics 53/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
single (scalar) and two species models
continuous and discrete time
nonlinearity, i.e density dependence
equilibria and local stability
harvesting, i.e. control
Stuart Townley Math Biol - Basics 53/ 54
Overview of week aheadSingle Population Models
Coupled population modelsSummary
Overview of week
X Lecture 1: Mathematical biology for one and two dimensionalmodels
Lecture 2: Population Projection Models - A Lecture by“Crowd Sourcing” Built into this topic is a group researchproject - Presentations on Friday
Lecture 3: A Feedback Control Approach to NonlinearPopulation Dynamics
Lecture 4: Diffusion Driven Instability and links to SwitchedSystems
Lecture 5: Further and future topics in “Systems Theory forMathematical Biology”
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