existence of periodic travelling waves solutions in predator
TRANSCRIPT
Existence of periodic travelling waves solutions in predator
prey model with diusion
Radouane YAFIA∗†, M. A. Aziz-Alaoui‡
July 19, 2012
AbstractThis paper deals with the qualitative analysis of the travelling waves solutions of a reaction
diusion model that refers to the competition between the predator and prey with modiedLeslie-Gower and Holling type II schemes. The well posedeness of the problem is proved.We establish sucient conditions for the asymptotic stability of the unique nontrivial posi-tive steady state of the model by analyzing roots of the forth degree exponential polynomialcharacteristic equation. We also prove the existence of a Hopf bifurcation which leads toperiodic oscillating travelling waves by considering the diusion coecient as a parameter ofbifurcation. Numerical simulations are given to illustrate the analytical study.
Keywords :Predator prey model, diusion, stability, periodic travelling waves, Hopf bifurcation
1 Introduction
The simplest reaction diusion models for cyclic populations involve two interacting species, withdensities u and v:
∂∂tu(t, x) = Du
∂2
∂x2 u(t, x) + f(u, v)
∂dtv(t, x) = Dv
∂2
∂x2 v(t, x) + g(u, v)(1.1)
where f(u, v) and g(u, v) model the local activity (absence of diusion), the densities u and v mayrepresent the predator and prey, host and parasite, herbivore and grazer, etc. Here x is spatial co-ordinate and t denotes time. Our focus on cyclic populations means that we assume that the localdynamics f and g are such that the spatially uniform equations du
dt = f(u, v); dvdt = g(u, v) have
a stable periodic solution (limit cycle), which oscillates around the unstable coexistence steadystate. The theory of periodic travelling waves is essentially the same for models with three or moreinteracting species.The model under study is given by a reaction diusion system based on the predator prey inter-action species with modied Leslie-Gower and Holling type II schemes (see, [1]):
∂dtu(t, x) = d1
∂2
∂x2 u(t, x) + u(t, x)(r1 − b1u(t, x)− a1v(t,x)
u(t,x)+k1
),
∂dtv(t, x) = d2
∂2
∂x2 v(t, x) + v(t, x)(r2 − a2v(t,x)
u(t,x)+k2
),
(1.2)
∗Corresponding author: R. Yaa.†Ibn Zohr University, Polydisciplinary Faculty of Ouarzazate, B.P: 638, Ouarzazate, Morocco. [email protected]‡Laboratoire de Mathématiques Appliquées, 25 Rue Ph. Lebon, BP 540, 76058Le Havre Cedex, France.
1
where all parameters in (1.2) are positives. The functions u(t, x) and v(t, x) are densities of theprey and predator, respectively, d1 and d2 are the diusion coecients, r1 is the growth rate ofthe prey u, b1 measures the strength of the competition among individuals of species u, a1 is themaximal value which per capita reduction rate of u can attain, k1 (respectively; k2) measures theextent to which environment provides to prey u (respectively to predator v) r2 describes the growthrate of u and a2 has a similar meaning to a1.The rst model proposed in this optic is given by an ordinary dierential equations (see, [1]) andread as follows
dxdt =
(r1 − b1x− a1y
x+k1
)x,
dydt =
(r2 − a2y
x+k2
)y
(1.3)
with initial conditions x(0) > 0 and y(0) > 0.This two species food chain model describes a prey population x which serves as food for a predatory.The delayed model of (1.3) (see, [8]) is given by a system of two delayed dierential equations asfollows:
dx(t)dt =
(r1 − b1x(t)− a1y(t)
x(t)+k1
)x(t),
dy(t)dt =
(r2 − a2y(t−τ)
x(t−τ)+k2
)y(t)
(1.4)
for all t > 0. Here, the discrete delay τ > 0 has been incorporated in the negative feedback of thepredator's density.The notion of global stability is studied by many other authors in the predator-prey systems withdelay [2, 17]. In [1], authors studied the boundeness and global stability of system (1.3)and in [8]authors studied the global stability and persistence of the delayed system (1.4) by using liapunovfunctional.The existence of periodic solutions and their stability are studied in [18, 19], by considering thedelay as a parameter of bifurcation.The spatio-temporal predator prey model without modication is given by (see Huang et al. [6])
∂dtu(t, x) = d1
∂2
∂x2 u(t, x) + u(t, x)(r1 − 1
K u(t, x)−B v(t,x)Eu(t,x)+1
),
∂dtv(t, x) = d2
∂2
∂x2 v(t, x) + v(t, x)(r2 − Cv(t, x) + D u(t,x)
Eu(t,x)+1
),
(1.5)
where all parameters in (1.5) are positives, for the meaning of all constants in (1.5) see [6], in whichauthors studied the existence of travelling waves of system (1.5). The method used to prove theresults are shooting argument in R4 together with a lyapunov function and Lasalle's InvariancePrinciple.Travelling waves have been observed in nature in many cyclic animals see [3, 4, 7, 12, 15], otherauthors are interested in the study of perioc travelling waves see [13, 16] and references therein.
Our goal in this paper is to study the dynamics of the resulting travelling waves system ofequation (1.2). We prove the well posedeness of the problem and asymptotic stability of the nontrivial steady state with respect to the diusion coecient. We establish the existence of limitcycle via Hopf bifurcation theorem, by using the diusion coecient as a parameter of bifurcation.The current work is organized as follows: In section 2, the well posedeness of the problem is proved.Section (3) is devoted to the existence and stability of the steady states of the model. In section 4,we prove the existence of limit cycle (periodic travelling waves) of the model. In the end we givesome numerical simulations and discussions.
2
2 Well posedeness of the problem
The reaction diusion system we focus on here reads as,∂dtu(t, x) = d1
∂2
∂x2 u(t, x) + u(t, x)(r1 − b1u(t, x)− a1v(t,x)
u(t,x)+k1
),
∂dtv(t, x) = d2
∂2
∂x2 v(t, x) + v(t, x)(r2 − a2v(t,x)
u(t,x)+k2
),
(2.1)
For further simplication, takingu∗(r1t,
√r1d2
x) = b1r1
u(t, x), v∗(r1t,√
r1d2
x) = a2b2r1r2
v(t, x), d = d1d2, a = a1r2
a2r1, b = r2
r1, e1 = b1k1
r1,
e2 = b1k2r1
, t′ = r1t, x′ =√
r1d2
x
and dropping the stars on u, v and the primes on x, t for convenience, we obtain the followingsystem
∂dtu(t, x) = d ∂2
∂x2 u(t, x) + u(t, x) (1− u(t, x))− au(t,x)v(t,x)u(t,x)+e1
,
∂dtv(t, x) = ∂2
∂x2 v(t, x) + bv(t, x)(1− v(t,x)
u(t,x)+e2
).
(2.2)
We will show that the reaction diusion system (2.2) generates a dynamical system and it isbiologically well posed on suitable Banach space.Let us set F = (F1, F2), U = (u, v) and D = diag[d, 1], where
F1(u, v) = u(1− u)− auv
u + e1
F2(u, v) = bv(1− v
u + e2)
Henceforth, considering also a boundary conditions and system (2.2) can be written as∂∂tU(t, x) = D∆U(t, x) + F (U), x ∈ Ω, t > 0
∂∂nU(t, x) = 0, x ∈ ∂Ω, t > 0
U(0, x) = ϕ(x), x ∈ Ω
(2.3)
Let X be the Banach space X1 ×X2, where X1 = X2 = C(Ω).The norm is dened by
|ϕ| = |ϕ1|+ |ϕ2|.
Let A0u and A0
v be the dierential operators A0uu = d1∆u
A0vv = d2∆v,
(2.4)
dened on the domain D(A0u) and D(A0
v) respectively, where
D(A0u) =
u ∈ C2(Ω) ∩ C1(Ω) : A0
uu ∈ C(Ω),∂u
∂n(x) = 0, x ∈ ∂Ω
D(A0v) =
v ∈ C2(Ω) ∩ C1(Ω) : A0
vv ∈ C(Ω),∂v
∂n(x) = 0, x ∈ ∂Ω
.
3
The closures Au of A0u, and Av of A0
u in Xi (i = 1, 2) generate analytic semigroups of boundedlinear operators Tu(t) and Tv(t) for t ≥ 0 such that u(t) = Tu(t)ϕ1
v(t) = Tv(t)ϕ2
(2.5)
are solutions of the following abstract linear dierential equations in Xi u′(t) = Auu(t)
v′(t) = Avv(t)(2.6)
An additional property of the semigroup is that for each t > 0, Tu(t) and Tv(t) are compactoperators. In the language of partial dierential equations u(t, x) = [Tu(t)ϕ1](x)
v(t, x) = [Tv(t)ϕ2](x)(2.7)
are classical solutions of the initial boundary value problem (2.3) with F1 = F2 = 0.Let
T (t) : X −→ X
dened byT (t) = Tu(t)× Tv(t).
Then T (t) is a semigroup of operators on X generated by the operator A = Au ×Av dened on
D(A) = D(Au)×D(Av)
andU(t, x) = [T (t)ϕ](x)
is the solution of the following linear system∂∂tU(t, x) = D∆U(t, x), x ∈ Ω, t > 0
∂∂nU(t, x) = 0, x ∈ ∂Ω, t > 0
U(0, 0) = ϕ(x), x ∈ Ω
(2.8)
Observe that the nonlinear term F is twice continuously dierentiable on U . Therefore, we candene the map [F ∗(ϕ)](x) = F (ϕ(x)) which maps X into itself and equation (2.3) can be viewedas an abstract ordinary dierential equation in X given by z′(t) = Az(t) + F ∗(z(t))
z(0) = ϕ(2.9)
While a solution z(t) of (2.9) can be obtained under restriction that ϕ ∈ D(A), a mild solution canbe obtained for every ϕ ∈ X by requiring only that z(t) is a continuous solution of the followingintegral equation
z(t) = T (t)ϕ +∫ t
0
T (t− s)F ∗(z(s))ds, t ∈ [0, α), (2.10)
4
where α = α(ϕ) ≤ ∞. Restricting our attention to function ϕ in the set :
XΛ =ϕ ∈ X : ϕ(x) ∈ Λ, x ∈ Ω
where
Λ =U = (u, v) ∈ R2 : u ≥ 0, v ≥ 0
,
and taking into account the denition of the function Fi i = 1, 2, we obtain that F1(0, v) = 0 andF2(u, 0) = 0 for U ∈ Λ. Thus, Corollary 3.2, p. 129 in [14] implies that the Nagumo condition forthe positive invariance of Λ is satised, i.e.,
limh→0
h−1dist(Λ, U + hF (U)) = 0, U ∈ Λ. (2.11)
On the other hand, a direct application of the strong parabolic maximum principle can be used toshow that the linear semigroup T (t) leaves XΛ positively invariant, i.e.,
T (t)XΛ ⊂ XΛ, t ≥ 0. (2.12)
Finally, conditions (2.11) and (2.12) together allow us to apply Theorem 3.1, p. 127 in [14], andwe obtain the following result.
Theorem 2.1. For each ϕ ∈ XΛ, (2.2) has a unique mild solution z(t) = z(ϕ, t) ∈ XΛ anda classical solution U(t, x) = [z(t)](x). Moreover, the set XΛ is positively invariant under owΨt(ϕ) = z(ϕ, t) induced by (2.2)
3 steady states and stability
Consider again system (2.2), there are several reasonable parameters restrictions.First, we require that 0 < d ≤ 1, which indicates that the prey population does not disperse fasterthan the predator.We also require that ae2 < e1, which ensures that system (2.2) has a positive equilibrium pointcorresponding to constant coexistence of the two species. System (2.2) has forth equilibrium points:E0 = (0, 0), E1 = (1, 0), E2 = (0, e2), E∗ = (u∗, v∗) which are equilibria of the correspondingordinary dierential equation system (2.2) without diusion, where
u∗ =1− a− e1 +
√(a + e1 − 1)2 + 4(e1 − ae2)
2, (3.1)
andv∗ = u∗ + e2. (3.2)
Without diusion , the equilibrium point E0 corresponding to the absence of both species isunstable, E1 corresponding to the prey at the environment carrying capacity in the absence ofpredators is a saddle point. If ae2 < e1, E2 corresponding to the predator at the environmentcarrying capacity in the absence of prey is also a saddle point and E∗ corresponding to coexistenceof the two species is asymptotically stable if p(u∗) > 0 and unstable if p(u∗) < 0, where
p(x) = 2x2 + (b + e1 − 1)x + be1. (3.3)
Furthermore, E∗ is asymptotically stable if b + e1 − 1 ≥ 0 or 0 < u∗ < α1 or α2 < u∗ < 1 and it isunstable if b+ e1−1 > 0 and α1 < u∗ < α2, where α1 and α2 are the roots of the polynomial p(x):
α1,2 =1− b− e1 ±
√(b + e1 − 1)2 + 8be1
4.
5
Particulary, if e1 − 1 ≥ 0, E∗ is globally asymptotically stable. If b + e1 − 1 ≥ 0 the system has nolimit cycle.In order to establish the existence of travelling waves solutions of system (2.2), we assume that thesolutions have the spacial form u(t, x) = u(x + ct) and v(t, x) = v(x + ct), where s = x + ct, and cis the wave speed. Then the system (2.2) becomes
cu′ = du′′ + u (1− u)− auvu+e1
,
cv′ = v′′ + bv(1− v
u+e2
),
(3.4)
where ′ denotes the dierentiation with respect to the travelling wave variable s.Rewrite (3.4) as a system of rst order ordinary dierential equations in R4:
u′ = w,
v′ = z,
w′ = cdw + 1
du(u− 1) + ad
uvu+e1
,
z′ = cz − bv + bv2
u+e2.
(3.5)
System (3.5) has forth equilibrium points E0 = (0, 0, 0, 0), E1 = (1, 0, 0, 0), E2 = (0, e2, 0, 0) andE∗ = (u∗, v∗, 0, 0), where u∗ and v∗ are given in (3.1) and (3.2) .The linearized system of the system (3.5) around an equilibrium point (f, g, 0, 0) is given by:
u′ = w,
v′ = z,
w′ = cdw +
(1d (2f − 1) + a
dge1
(f+e1)2
)u + a
df
f+e1v,
z′ = cz − b g2
(f+e2)2u +
(−b + 2b g
f+e2
)v.
(3.6)
and the characteristic matrix is reads as
0 0 1 0
0 0 0 1
1d (2f − 1) + a
dge1
(f+e1)2ad
ff+e1
cd 0
−b g2
(f+e2)2−b + 2b g
f+e20 c
and the associated characteristic equation is given by
Q(λ, d) = λ4 − (c + γ)λ3 + (cγ − α− υ)λ2 + (γυ + αc)λ + αυ − βµ, (3.7)
whereα =
1d
(2f − 1) +a
d
ge1
(f + e1)2,
β =a
d
f
f + e1,
6
γ =c
d,
µ = −bg2
(f + e2)2,
υ = −b + 2bg
f + e2.
For βµ = 0, (3.7) becomes
Q(λ, d) = λ4 − (c + γ)λ3 + (cγ − α− υ)λ2 + (γυ + αc)λ + αυ
= (λ2 − cλ− υ)(λ2 − γλ− α).
Remark 3.1. βµ = 0, implies that β = 0 or µ = 0.If β = 0, we have f = 0 and from the equation (3.7) we deduce the stability of the equilibrium pointE2.If µ = 0, we have g = 0 and from the equation (3.7) we deduce the stability of the equilibriumpoints E0 and E1.
At the equilibrium point (f, g, 0, 0), with (f, g) = (u∗, v∗) we have
f − 1 +a
d
fg
f + e1= 0 and g = f + e2
and we obtain thatα =
1df − a
d
fg
(f + e1)2,
β =a
d
f
f + e1,
γ =c
d,
µ = −b,
υ = b,
and the characteristic equation is as follows
Q(λ, d) = λ4−c(1+1d)λ3+(
c2
d−b−f
d(1− ag
(f + e1)2))λ2+
c
d(b+f(1− ag
(f + e1)2))λ+
bf
d(
a
f + e1+1− ag
(f + e1)2).
(3.8)Note that
p(f)f + e1
= b + f(1− ag
(f + e1)2,
with p(x) is the polynomial dened in (3.3).For more simplication, denote by ρ = ρ(f) = af
f+e1− b, θ = θ(c) = c2 + b(1− d) and σ = σ(f) =
p(f)f+e1
.Then
Q(λ, d) = λ4 − c(1 +1d)λ3 +
θ − σ
dλ2 +
cσ
dλ +
b(ρ + σ)d
(3.9)
Theorem 3.1. 1) If c > 0, σ < 0 and ρ+σ > 0, the equilibrium point (f, g, 0, 0) is a saddle point.2) If c < 0, σ < 0, −ρ < σ < 0 and X1 < 1 + d < X2, the equilibrium point (f, g, 0, 0) isasymptotically stable and unstable if 1 + d > X2. Where
X1 =(c2 + 2b)σ +
√(c2 + 2b)2σ2 + 4σ2bρ
−2bρ< 0, (3.10)
X2 =(c2 + 2b)σ −
√(c2 + 2b)2σ2 + 4σ2bρ
−2bρ> 0. (3.11)
7
Proof. 1) From the Hurwitz criteria, we deduce that the equilibrium point (f, g, 0, 0) is a saddlepoint.2) Let ai, i = 1, 2, 3, 4 denote the coecients of the polynomial Q dened in equation (3.9).As c < 0, σ < 0, −ρ < σ < 0, then a1 > 0, a2 > 0 and a4 > 0. To apply the Hurwitz criteria weneed that a3 > 0.Suppose now X1 < 1 + d < X2, where Xi, i = 1, 2 dened in (3.10) and (3.11) are the roots of thefollowing polynomial
D(x) = −bρx2 − (c2 + 2b)σx + σ2. (3.12)Which imply that D(1 + d) > 0 and by a simple calculus, we obtain that a3 > 0. From Hurwitzcriteria we deduce the result.To obtain the switch of stability, one needs to nd the purely imaginary roots of (3.9).Let λ = ik a root of (3.9), then
Q(ik, d) = 0,
imply that
(ik)4 − c(1 +1d)λ3 +
θ − σ
d(ik)2 +
cσ
d(ik) +
b(ρ + σ)d
.
Separating the real and imaginary parts, we obtain
k4 − θ − σ
dk2 + b
ρ + σ
d= 0, (3.13)
andc(1 +
1d)k3 +
cσ
dk = 0. (3.14)
From equation (3.14), we havek2 = − σ
1 + d> 0.
By replacing the last quantity in equation (3.1), we obtain that
D(1 + d) = 0
and the equilibrium point (f, g, 0, 0) is unstable for 1 + d > X2 and (3.9) has a conjugate purelyimaginary roots at d = d0 = X2 − 1.
4 Existence of limit cycle
In this section, we consider the diusion coecient d as a parameter of bifurcation and we provethat the system (3.5) has a limit cycle as d passes trough the critical value d0 via Hopf bifurcationtheorem.
Theorem 4.1. Assume c < 0, σ < 0, −ρ < σ < 0 and X1 < 1 + d < X2. Then, there existsε0 > 0 such that, for each 0 ≤ ε < ε0, equation (3.5) has a family of periodic solutions pl(ε) withperiod Tl = Tl(ε), for the parameter values d = d(ε) such that pl(0) = (f, g, 0, 0), Tl(0) = 2π
k andd(0) = d0 = X2 − 1, where X2 is given in equation (3.11).
Proof. From system (3.5) the nonlinear term is given by
H(d, X) =
0
0
1d (u2 + u∗(u∗ − 1)) + a
d(u+u∗)(v+v∗)
u+u∗+e1−(
1d (2f − 1) + a
dge1
(f+e1)2
)u− a
df
f+e1v
−bv∗ + b(v+v∗)2
u+u∗+e2+ b g2
(f+e2)2u− 2b g
f+e2v
(4.1)
8
where X = (u, v, w, z) ∈ R4.Then, we have
H(d, 0) = 0 and∂
∂XH(d, 0) = 0.
Now, we need to verify the transversality condition. From equation (3.9), we have
Q(ik, d0) = 0 and∂
∂λQ(ik, d0) 6= 0
As the function Q is continuous and from the implicit functional Theorem, we have Q(λ(d), d) =0 for d in a neighborhood of d0 and
λ′(d)|d=d0 = − −k4 − ick3 − bk2
−cσ − 4d0ik3 + 3cd0k2 + 2(c2 + 2b− bd0 − σ)ik. (4.2)
ThereforeRe(λ′(d)|d=d0) = k4 M
N,
whereM = −4c(d0 + 1)(k2 + b)− 2c((1− d0)k2 + c2 + b(1− d0)),
N = (3ack2 − cσ)2 + (4(a− 1)k3 − 2(c2 + 2b− b(d0 + 1)− σ)k)2.
Re(λ′(d)|d=d0) > 0.
From the Hopf bifurcation theorem for ordinary dierential equations [9, 11], we deduce the result
5 Numerical simulations and discussions
With Matlab software we illustrate our result by some numerical simulations. The method usedto compute the travelling waves solutions is to use BVP solver with projection conditions and aphase condition with the following parameters values a = 0.5, b = 0.25, c = 2, e1 = 2, e2 = 2.5 and0 < d < 1 From gure 1, we observe that for small density prey the density of predator decreases
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−1
0
1
time t
u, v
u predatorv prey
Figure 1: Travelling wave solution connecting the steady states (1, 0) and (u∗, v∗), predator (blueline) and prey (green line)
9
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−4
−3
−2
−1
0
1
2
3
4
x 1012
−1
−0.5
0
0.5
1
time t
u, v
u predatorv prey
Figure 2: An illustration of the existence of periodic travelling waves connecting the steady states(1, 0) and (u∗, v∗) for d = d0 predator (blue line) and prey (green line)
to reach the equilibrium state u∗ and the prey density of prey increases to reach the equilibriumv∗ for d < d0. But in gure 2, the two population coexist but they oscillate periodically aroundthe nontrivial steady state (u∗, v∗) for d = d0 and the two population can survive together forlong time (because the prey may exists with a high density (1012) and there is no extinction of thetwo population). In Fig. 3, an illustration of the periodic travelling waves solution with respectto the time and space variables and Neumann boundary conditions. The solutions of the reactiondiusion system are represented by a surface, for each xed x the solutions are represented by lineswhich correspond with travelling waves in dimension two. We observe that, the periodicity of theinvasion of the predators imply the periodicity of the prey.In this paper, we show the existence of periodic travelling waves solutions via Hopf bifurcationTheorem by considering the diusion coecient as a parameter of bifurcation. A numerical simu-lations are given to illustrate the theoretical study of the reaction diusion system modelling theinvasion of the prey by the predator species. The predation is an established case of cycling in preyspecies. Here, the ability of predation to explain periodic travelling waves in prey population (seeFig. 2), which have recently been found in a number of spatiotemporal eld studies. The natureof periodic waves in this systems, and the way in which they can be generated by the invasionof predators into a prey population. A theoretical calculation that predicts, as a function of oneparameter ratio, whether such an invasion will leads to a periodic travelling waves that would beobserved. The result gives an insight into the types of predator prey systems in which one wouldexpects to see periodic travelling waves following an invasion by predators.
5.1 Conclusion
In this work, a spatio-temporal system reaction-diusion modelling predator-prey population withmodied Leslie-Gower and Holling type-II functional response is studied. By using the semigrouptheory, we showed the well-posedeness of the problem and the positivity of solutions. In Theorem3.1, the conditions of change of stability of possible steady states are given. By considering thediusion coecient as a parameter of bifurcation and by applying the Hopf bifurcation Theorem,we prove that there exists a critical value of diusion parameter at which a small amplitude periodicsolution in R4 which corresponds to a small amplitude travelling wave solution connecting the twoequilibrium points occurs (Theorem 4.1).Acknowledgment: We would like to thank the referees and the editor for their careful reading ofthe original manuscript and their many valuable comments and suggestions that greatly improved
10
Figure 3: Periodic travelling waves with respect to the time and space variables
the presentation of this work. We also thank the Prof. A. Talibi for their discussions and remarks.
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