dynamics of a plant-herbivore model with …kuang/paper/gypsymoth.pdf · dynamics of a...

August 14, 2007 17:6 Journal of Biological Dynamics DiscreteModel˙final

Journal of Biological DynamicsVol. 00, No. 00, Month-Month 200x, 1–12

Dynamics of a plant-herbivore model with applications to gypsy moth outbreaks

Yun Kang, Dieter Armbruster and Yang Kuang(Received 00 Month 200x; revised 00 Month 200x; in final form 00 Month 200x)

We formulate a novel host parasite model to study the dynamics of the outbreak of the gypsy moths. Assuming a Ricker dynamics forthe host population and an infestation that takes place after the growth limitations take effect in the plant dynamics, a two dimensionaldiscrete model of the leaf mass and the gypsy moths mass is constructed. The parameter space is determined by the growth rate of thehost population and a parameter describing the damage done by a gypsy moth. Bifurcation curves in that parameter space are presented.Bistability and a crises of a strange attractor suggest two control strategies: Reducing the population of the gypsy moths under somethreshold or increasing the growth rate of the plant leaves.

Keywords: Bifurcation, chaos, gypsy moth, conservation of biomass production, control strategies.

AMS Subject Classification: 34K20, 92C50, 92D25.

1. Introduction

The usual framework for discrete-generation host-parasite models have the form

Pn+1 = λPnf(Pn,Hn), (1)

Hn+1 = cλPn [1− f(Pn,Hn)] , (2)

where P and H are the population size of the host (a plant) and parasite (a herbivore) in successivegenerations n and n + 1 respectively, λ is the host finite rate of increase (λ = er where r is the intrinsicrate of increase) in the absence of the parasites, c is the biomass conversion constant, and f is the functiondefining the fractional survival of hosts from parasitism. Throughout the rest of this paper,n takes thevalue of nonnegative integers.

The simplest version of this model is that of Nicholson (1933) and Nicholson and Bailey (1935) whoexplored in depth a model in which the proportion of hosts escaping parasitism is given by the zero termof the Poisson distribution namely,

f(Pn,Hn) = e−aHn , (3)

where aHn are the mean encounters per host. Thus, 1−e−aHn is the probability of a host will be attacked.Substituting into the equations (1) and (2) gives:

Pn+1 = λPne−aHn , (4)

Hn+1 = cλPn

[1− e−aHn

]. (5)

When λ > 1, the Nicholson-Bailey model has a positive equilibrium and it is unstable (p81 in Edelstein-Keshet 2005) with the slightest perturbation leading to oscillations of rapidly increasing amplitude.

Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804, USA. E-mail:[email protected],[email protected], [email protected]

Journal of Biological DynamicsISSN: 1751-3758 print/ISSN 1751-3766 online c© 200x Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/1751375YYxxxxxxxx


2 Dynamics of a plant-herbivore model

In a long term population dynamics, density dependent regulation (other than parasitism) acting onthe host population is ubiquitous. Such a mechanism has been introduced by Maynard Smith and Slatkin(1973) for an age class extension of the Nicholson-Bailey model, and for the Nicholson-Bailey model byBeddington et al. (1975) who adopted a discrete version of the logistic equations (May 1973)

Pn+1 = λPne

0@−

rPn

Pmax− aHn

1A

, (6)

Hn+1 = cλPn

(1− e−aHn

), (7)

where Pmax is the so-called environment imposed “carrying capacity” for the host in the absence of theparasite. What was an unstable positive equilibrium when no density dependent is assumed for hostpopulation growth becomes locally stable for a large set of parameter values (Beddington et al. 1975.See also p84 in Edelstein-Keshet 2005). For other parameter values, the model can generate attractors ofvarious complexity, ranging from a set of limit cycles of various periods to strange and chaotic ones.

Observe that the structure of equations (6) and (7) implies that the host density-dependence acts ata particular time in their life cycle in relation to the stage attacked by the parasites. The Hn herbivoressearch for Pn hosts before the density dependent growth regulation kicks in. Hence the next generationof herbivores depends on Pn, the initial host population prior to parasitism. This leads to the followinggeneral form:

Pn+1 = λPng(Pn)f(Hn), (8)

Hn+1 = cλPn [1− f(Hn)] , (9)

where f(Hn) represents the fraction of hosts surviving parasitism, and the host density dependence takesthe form g(Pn) = e(1− rPn

Pm).

In reality, at least in cases of many observable plant-herbivore (host-parasite) interactions, herbivores(parasites) are not attracted to and attack plants until their leaves form good canopies. Moreover, thefeeding process continues throughout most periods of the growing season. This argues for plant-herbivore(host-parasite) models where the act of herbivores (parasites) occurs after the density dependent growthregulation of the host takes place. Intuitively, the herbivore population depends on the density regulatedplant population level. We propose the following discrete models for such plant-herbivore interactions:

Pn+1 = λPng(Pn)f(Hn), (10)

Hn+1 = cλPng(Pn) [1− f(Hn)] . (11)

Here we assume that the grown but lost plant biomass is fully converted to the biomass of herbivore. Inother words, model (10)-(11) observes the conservation of biomass production. This basic conservationlaw is often ignored in existing population models without due justifications. The above mentioned dra-matic difference of the dynamics of the modified Nicholson-Bailey model (6)-(7) from that of the originalNicholson-Bailey model makes us wonder if the dynamics of the model (10)-(11) is comparable to that ofmodel (8)-(9). This question will be pursued below in the context of a specific plant-herbivore interaction.

2. A gypsy-moth model and its boundary dynamics

In the following, we would like to apply the above model framework to the interaction between a plantpopulation and its parasite. Specifically, we would like to study deciduous plants and a herbivore, thegypsy moth, which is a notorious forest pest in North Central United States whose outbreaks are almostperiodic and cause significant damage to the infested forests. We will model the plants as well as gypsy


Yun Kang, Dieter Armbruster and Yang Kuang 3

moths through their nutrient content changes. We assume that the soil acts as an unlimited reservoirof nutrients. A plant takes up the nutrient from the soil, stores the nutrient in its stem, bark, twig andleaves. During spring, leaves emerge in a deciduous forest. Gypsy moths hatch from the egg mass afterbud-break, and start to feed on new leaves. At the end of the season n, gypsy moths lie eggs and die.More specifically, we make the following assumptions:

A1: Pn represents the nutrient in the plant after an attack by the gypsy moths but before its defo-liation. Hn represents the nutrient contained in the gypsy moths before they die at the end of the season n.

A2: Without the herbivore, the nutrients stored in the plants follows the dynamics of the Ricker model(Ricker 1954),

Pn+1 = Pner

241−

Pn

Pmax

35

(12)

with a constant growth rate r and pant carrying capacity Pmax. The Ricker dynamics determines theamount of new leaves available for consumption for the gypsy moth larvae.

A3: We assume that the gypsy moths search for food randomly. The leaf area consumed by the gypsymoth is measured by the parameter a, i.e., a is a constant that correlates to the total amount of thenutrients that a gypsy moth consumes during its larvae stages.

After the attacks by the gypsy moths, the nutrient in the plants is reduced to e−aH(n) fraction of thelevel when no gypsy moths present. Hence,

Pn+1 = Pner

241−

Pn

Pmax

35−aHn

. (13)

After egg hatching in the early spring, larvae start to feed on the young leaves. The amount of decreasednutrient in the plants is converted to the nutrient of the gypsy moth. Mathematically, the conversionparameter can be scaled out by a simple change of variable Hn/c → Hn. Hence, for convenience, weassume below that the biomass conversion parameter is 1. Therefore, at the end of the season n, we have

Hn+1 = Pner

241−

Pn

Pmax

35 [

1− e−aHn]. (14)

There are three parameters in our system r, a and Pmax. We can scale Pmax away by setting xn =Pn

Pmax, yn =

Hn

Pmax, a → aPmax. This yields the following simplified system:

xn+1 = xner[1−xn]−ayn , (15)

yn+1 = xn

[1− e−ayn

]er[1−xn]. (16)

It is easy to see that if x0 = 0 (y0 = 0), then xn = 0 (yn = 0) for n > 0. We assume that a > 0 andr > 0. Observe that xn+1 + yn+1 = xner(1−xn ≤ er−1/r. We thus have the following proposition.

Proposition 2.1 Assume that a > 0, r > 0, x0 > 0 and y0 > 0, then xn > 0 and yn > 0 for all n > 0. Inaddition we have maxn∈Z+xn, yn ≤ er−1/r for n > 0.

The Ricker dynamics is a prototype of an ecological model describing non-overlapping populations witha one hump dynamics. The one hump dynamics indicates an optimal population size - deviations from



that size lead to reduced populations in the next generation. This reflects any kind of limitations of thegrowth of a population - finite food, water or energy supply etc.

Guckenheimer (1979) showed that such maps are topologically equivalent to the logistic map and henceshow a period doubling sequence to a chaotic behavior, i.e. sensitive dependence on initial conditions anda dense orbit. Figure 1 shows the attracting sets as a function of the growth rate r. It is known that atr = 2 the system undergoes period doubling to a period 2 orbit, at r = 2.5, its period doubles again to aperiod 4, etc. The r value sequence for period doubling sequence converges to rc = 2.6924 at which timewe have chaos.

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6Bifurcation Diagram of Ricker Model in r

x[n]

r

x0=0.2, N=200

Figure 1. Bifurcation Diagram of Ricker’s Model

The system (15)-(16) has the two boundary equilibria (0, 0) and (1, 0) and possibly multiple interiorequilibria depending on the parameter values of r and a.

At the boundary equilibrium point (0, 0), the Jacobian matrix takes the form of

J |(0,0) =

[er 0

0 0

](17)

with eigenvalues λ1 = er ≥ 1 and λ2 = 0 < 1. Hence, the origin is a saddle which is stable on the y-axisand unstable on the x-axis.

At the boundary equilibrium point (1, 0), the Jacobian matrix is

J |(1,0) =

[1− r −a

0 a

](18)

with eigenvalues λ1 = 1 − r and λ2 = a. As a result, we have two codimension one bifurcating from thisequilibrium:

• Near r = 2 and a 6= 1, the Taylor expansion of equations (15) on the invariant manifold y = 0 gives

un+1 = (−1− γ)un +2u3

n

3(19)

where γ = r − 2 and un = xn − 1. Equation (19) satisfies the necessary and sufficient conditions forperiod doubling [11] consistent with the Ricker dynamics.



• Near a = 1 and r 6= 0, 2, we can perform a center manifold reduction to the reduced equation

vn+1 = (1 + α)vn − (α + γ)v2n

2− v3

n

3(20)

where γ = r − 2, α = a− 1 and vn = yn. For γ 6= 0, Equation (20) satisfies the necessary and sufficientconditions for a transcritical bifurcation. For α = γ = 0 we have a pitchfork bifurcation.

• We can unfold the transcritical bifurcation to obtain a saddle node bifurcations at

v =√−3α, γ = −4

3√−3α− α, (21)

v = −√−3α, γ =43√−3α− α. (22)

Notice that the saddle node bifurcation (22) lies in the second quadrant and hence is biologically unin-teresting and that the saddle node bifurcation (21) exists only for α < 0.

• When r = 2 and a = 1, we have the most degenerate case with eigenvalues λ1 = −1 and λ2 = 1.

3. Interior equilibria

Let E = (x∗, y∗) be an interior equilibrium of model (15)-(16). From (15) we obtain x∗ = 1 − ay∗

rand

from (16) we obtain x∗ = y∗/(eay∗ − 1) (since ay∗ = r(1− x∗)). Let

f1(y) = 1− ay

rand f2(y) =

y

eay − 1.

Then the interior equilibria are the intersection points of these two functions in the first quadrant Q+.Clearly, f1(y) is a decreasing linear function. It is easy to show that f2(y) is also decreasing. In addition,

we have limy→0 f2(y) = 1/a. In addition, limy→0 f ′2(y) = −1/2. Observe that if a = 1, r = 2, f1(y) andf2(y) are tangent at the boundary equilibrium (1, 0) with a slope of −1/2. Straightforward computationshows that f ′′2 (y) ≥ 0 and f ′2(y) ∈ (−1/2, 0), for y > 0. Hence the intersection of f1(y) and f2(y) in thefirst quadrant Q+ has 0 (Figure 2(a)), 1 (Figure 2(b)) or 2 (Figure 2(c)) interior equilibria.

1.41

0.6

0.80.6

1

0.4

0.2

00.2

y

1.2

1.2

0.4

0

0.8

red--x=1-ay/r

blue--x=y/(exp(ay)-1)

(a) No interior equilibrium: r = 1.2,a =0.8

1

0.8

0.6

0.4

0.2

0

y

21.510.50

red:x=1-ay/r

blue:x=y/(e^(ay)-1)

(b) One interior equilibrium: r = 0.98, a =2.6

0.50

y

2.5

0.8

0.6

21.50

1

0.4

1

0.2

red:x=1-ay/r

blue:x=y/(e^(ay)-1)

(c) Two interior equilibrium points: r =2.6,a = 0.98

Figure 2. Interior equilibria

Specifically, we have the following Proposition.

Proposition 3.1 For a > 1, model (15)− (16) has an unique positive equilibrium (see Figure 2(b)).



Proof : For a > 1, we see that f2(0) < f1(0). Moreover f ′2(y) ∈ (−1/2, 0), f ′′2 (y) ≥ 0, f ′1(y) = −a/r andf2(r/a) > f1(r/a) = 0. ¤

By the Jury test (p57 in Edelstein-Keshet 2005), we see that the interior equilibrium is locally asymp-totically stable if and only if

2 > 1 + det(J) = 1 + ax∗(1− rx∗)eay∗ > |tr(J)| = |1 + (a− r)x∗|.

3.1. Codimension one: Neimark-Sacker bifurcation

The condition for a Neimark-Sacker bifurcation are that λ · λ = 1 and λ is not a real number. We havethe following Theorem.

Theorem 3.2 A necessary condition for a Neimark-Sacker bifurcation to occur at an interior equilibriumpoint is

er[1−f1(r,a)] =r[1− f1(r, a)]

af1(r, a)+ 1 and |tr(Ji)| = |1− rf1 + af1| ≤ 2 (23)

or

er[1−f2(r,a)] =r[1− f2(r, a)]

af2(r, a)+ 1 and |tr(Ji)| = |1− rf2 + af2| ≤ 2 (24)

Here f1(r, a) and f2(r, a) are the x-components of the interior equilibria given by

f1(r, a) =−r2 + a− r +

√r4 + 2ar2 − 2r3 + a2 − 6ar + 5r2

2r(−r + a), (25)

f2(r, a) =−r2 + a− r −√r4 + 2ar2 − 2r3 + a2 − 6ar + 5r2

2r(−r + a). (26)

If (23) and (24) hold, then the solution sets for the two equations (23) and (24) are the bifurcation curvesfor a Neimark-Sacker bifurcation in the parameters space (a, r).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2nutritient modelling

Inse

ct/y

plant/x

Periodic Solution P=9a=2.1, r=2.7

Figure 3. Periodic Solution with period 9 for r = 2.7, a = 2.1



0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.341.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55nutritient modelling

Inse

ct

plant

a=1.5, r=2.45,N=10000

Figure 4. Dense orbit in phase space for r = 2.45, a = 1.5

Neimark-Sacker bifurcations generate dynamically invariant circles. As a result, we may find isolatedperiodic orbits as well as trajectories that cover the invariant circle densely. Figure 3 shows a period 9orbit while Figure 4 illustrates a dense orbit resembling an invariant circle.

There exists no bifurcation with codimension higher than 1 in the interior. The most degenerated caseoccurs at the boundary equilibrium (1, 0) which simultaneously undergoes a pitchfork and a period doublingbifurcation at (r, a) = (2, 1).

3.2. Bifurcation diagram in the (r, a)-parameter space

The codimension one bifurcations discussed before allow us to determine regions in parameter space r anda for which the dynamics of the system is topologically equivalent, Figure 5 shows the relevant (r, a) space.

Recall that the Ricker dynamics on the invariant manifold y = 0 is a unimodal map and shows a perioddoubling cascade to chaos as the parameter r is increased. In Figure 5, we indicate the stable equilibriumfor the Ricker dynamics through no shading for r < 2, stable periodic behavior of the Ricker dynamicsfor 2 ≤ r < rc through dark yellow, and the chaotic behavior including its periodic windows for r > rc bylight yellow.

In addition we show the following codimension one curves.

(i) The transcritical bifurcation at a = 1 in green.(ii) The saddle node bifurcation emanating from the pitchfork bifurcation at a = 1, r = 2 drawn in red.

Only the part that leads to a saddle node bifurcation in the first quadrant is shown.(iii) The Neimark Sacker bifurcation generated as a level set of the equations (23) and (24) in light blue.(iv) The blue line in the Figure 5 connects simulations shown as red diamonds representing values of r and

a from Table 1. The line approximately indicates the collapse of a strange attractor through a crisis(see below).

These curves divide the parameters space (r, a) into seven regions labeled one to seven in Figure 5. Oursimulation results suggest the following.

1). The boundary equilibrium (1, 0) is a globally stable attractor.2). The system has no interior equilibrium and the Ricker dynamics is globally attracting to either periodic

orbits or to chaotic orbits on the x-axis.3). The system has gained two interior equilibria, The one with the larger y-value is stable. The boundary

equilibrium is still stable with respect to perturbations into the y-direction.4). The stable interior equilibrium has lost stability through a Neimark Sacker bifurcation. We numerically



0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

value of r

valu

e of

a

Graph of Bifurcation Diagram

1

3

2

7

5

6

4

rc

Figure 5. Bifurcation Diagram of Parameters r and a

find stable periodic orbits (isolated and dense) and, as the values of r and a increase, strange attractors.5). The transition between region 4 and the region 5 is the numerically generated curve where the strange

attractor collapses in a crisis bifurcation. Inside region 5, there is no attractor in the interior of thephase space. All interior points are attracted to x-axis following the mostly chaotic Ricker dynamics.

6). The system has only one interior equilibrium which is unstable. There is a limit cycle around theunstable interior equilibrium. As the values of r and a increase, that limit cycle becomes unstable andforms a strange attractor. The transition between the region 6 and the region 7 is again a Neimark-Sacker bifurcation.

7). One stable interior equilibrium which appears to be the global attractor. The transition between theregions 5, 6, 7 and the regions 1, 2, 3, 4 is a transcritical bifurcation.

4. Main dynamical features

The discussion of the bifurcation curves in the parameter space (r, a) shows that for the same value of a,the dynamics of our model tend to be more stable as r decreases. Below we would like to highlight thefrequently observed bistability and chaos.

4.1. Bistability

Theorem 4.1 For a < 1 and r > 2, the period two orbit on the invariant manifold y = 0 is transversallystable.

Proof : The eigenvalue governing the local transverse stability of an otherwise attracting periodic orbit onthe invariant manifold is given by Πn=N

n=1 axner(1−xn) where the set xn denotes all iterations of a periodicorbit and N denotes its period. Alternatively, if the dynamics on the invariant manifold is chaotic, theeigenvalues becomes the Lyapunov number since N → ∞. However xner(1−xn) = xn+1 for the Rickerdynamics and hence the transverse eigenvalue (Lyapunov number) becomes Πn=N

n=1 axn. At the equilibriumx = 1 the eigenvalue becomes aN and therefore by assumption the equilibrium is attracting in the y-direction. For the period two orbit the eigenvalue for a = 1 becomes x1x2 = x2

1er(1−x1) which for x > 1

lies in the interval [0, 1]. Hence the period two orbit, whenever it exists is transversally stable. ¤



Numerically we find that any attractor on the invariant manifold is transversally stable and hence alocal attractor, if a is small enough.

0 20 40 60 80 100 1200.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8Time Series for Plant

Pla

nt

Time

a=0.98,r=2.45,x0=0.2,y0=0.25

(a) Time series for the Plants a = .98, r = 2.45 andinitial conditions x0 = 0.2, y0 = 0.25

0 20 40 60 80 100 1200.56

0.58

0.6

0.62

0.64

0.66

0.68Time Series for Plant

Pla

nt

Time

a=0.98,r=2.45

(b) Time series for the Plants a = .98, r = 2.45 andinitial conditions x0 = 0.57, y0 = 0.9

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

0.3

0.35Time Series for Insect

Inse

ct

Time

a=0.98,r=2.45,x0=0.2,y0=0.25

(c) Time series for the Insects a = .98, r = 2.45 andinitial conditions x0 = 0.2, y0 = 0.25. The forest pesteventually dies out.

0 20 40 60 80 100 1200.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99Time Series for Insect

Inse

ct

Time

a=0.98,r=2.45

(d) Time series for the Insects a = .98, r = 2.45 andinitial conditions x0 = 0.57, y0 = 0.9. The forest pesteventually reaches an endemic equilibrium.

Figure 6. Bistability in region 3

Theorem 4.1 implies that, anytime we have a stable dynamics in the interior of the phase space, anda small enough, we have bistability between that interior dynamics and the attractor, corresponding tothe value of r in the Ricker dynamics. This is true for region 3 where we have stable interior equilibriaand stable periodic orbits in the Ricker dynamics. Figure 6 a)-d)) shows an example of bistability betweena period two orbit on the x-axis and a stable interior equilibrium. It is also true for regions 6 and 7 asdetermined through numerical simulations listed in Table 1.

interior attractor vs. boundary attractor Value of r Value of aCase 1. limit cycle versus period two 2.6265 1.5Case 2. limit cycle versus period four 2.65 1.5Case 3. period 21 versus chaotic on x-axis 2.81 1.5Case 4. period 6 versus chaotic on x-axis 2.98 1.5

Table 1. Bistability Table

The Table 1 illustrates the different cases of bistable phenomena, when we fix the value of a = 1.5, then



vary the value of r from r < 2 up to r = 3.6. Figure 7 shows the associated phase space dynamics.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8Fixed pts or limit cycles

y va

lues

of f

ixed

pt

x values of fixed pt

(a) Interior and boundary attractor for case 1 in Table1.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


y va

lues

of f

ixed

pt


(b) Interior and boundary attractor for case 2 in Table1.

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2


y va

lues

of f

ixed

pt


(c) Interior and boundary attractor for case 3 in Table1.

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2


y va

lues

of f

ixed

pt


(d) Interior and boundary attractor for case 4 in Table1.

Figure 7. Bistability in regions 6 and 7

4.2. Strange attractor and its collapse

Simulations in regions 4 and 6 shows strange attractors - a typical picture is shown in Figure 8. The Table2 below is the result of a manual search for the stability boundary of the interior strange attractor. If thevalues of r or a are equal to or bigger than the values in this table, the strange attractor disappears andthe population of yn goes to zero.

a 0.7 0.8 0.85 0.9 0.95r 4.35 4.1 4.05 3.95 3.85a 1 1.5 2 2.5 4r 3.8 3.3 3.1 3.08 3.1

Table 2. Strange Attractors

The Ω-limit set of the initial conditions then will be the appropriate attractor on the x-axis. Figure 9gives an indication of how the strange attractor collapses. For a = 0.95, r = 3.8 we have bistability betweentwo strange attractors: an interior one and a boundary one. The shaded set is the interior strange attractor



while the streaks of iteration points outside are the stable manifold of the strange attractor on the x-axis.It is highly likely that in the empty space between those two sets of trajectories there is a periodic orbitof saddle type with high period. As the interior strange attractor grows further, it will collide with thatunstable periodic orbit, leading to a “leakage” of the strange attractor towards the x-axis.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6nutritient modelling

Inse

ct

plant

a=1.45,r=2.95,x0=.57,y0=.9

Figure 8. Strange Attractor:a = 1.45, r = 2.95

Figure 9. The interior strange attractor and the stable manifold of the boundary attractor a = 0.95, r = 3.8

5. Discussion

This paper models the outbreak of a gypsy moth infestation through nutrient transfer from plants to theherbivores. Our nonoverlapping 2 D model is controlled by the two parameters r, the nutrient up-takingrate by the plants, and a the amount of leaves eaten by the gypsy moths. Mathematical analysis and



simulations of this model may provide us with biological insights that may be used to devise controlstrategies to regulate the population of the gypsy moth. The following two strategies are the results of thedynamics discussed in the last sections.

(i) Exploiting bi-stability. For a small enough the system shows bistability between attractors withnonzero population sizes for the gypsy moth corresponding to an endemic situation, and attractors onthe x-axis corresponding to a dying out of the gypsy month. Hence, if the population of the gypsy mothinitially is small enough, it eventually will become extinct without further interference. This suggestscontrol actions to reduce the gypsy moth population by e.g. spraying of pesticides. We remark that thebistability of our system provides an alternative and possibly more plausible mechanism to that of theAllee effect employed by Tobin et al. (2006) to support the notion that a gypsy moth population can becontained if it is reduced below some threshold level. Allee effect is a theoretical hypothesis that oftenlacks solid mechanistical ground. Indeed, we may argue that this bistability may provides a possiblemechanism that explains the often observed Allee effect phenomenon for plant-gypsy interaction.

(ii) Exploiting the crisis of the strange attractor. For all values of a, as the parameter r becomeslarge enough, the interior dynamics becomes unstable and all trajectories starting in the interior even-tually approach the x axis, corresponding again to the extinction of the gypsy moth population. As rrepresents the growth rate of the plant species an increase could correspond to a natural occurrence ofhighly favorable growing conditions but also could be supported by fertilizing the plants. This seems toagree with the observation that gypsy moth outbreaks often take place following some draught yearsand when the plants suffer a period of sustained slow growth. Indeed, draught is regarded as the mainculprit of 2007 summer’s severe gypsy moth outbreak in Maryland, US [12].

It is well known that the modified Nicholson-Bailey model (6)-(7) can generate interior attractors inthe forms of a single equilibrium, limit cycle like set, cycles and eventually chaos. However, the modifiedNicholson-Bailey model (6)-(7) does not produce any interesting bistability dynamics or crisis of a strangeattractor which are important features provided by our model (15)-(16). In view of the fact that model(15)-(16) has only two free parameters, we suggest that it possessess better potential in generating usefulbiological insights and practical forest management policies.

Acknowledgments

The research of Dieter Armbruster is partially supported by NSF grant DMS-0604986, the research of YunKang is partially supported by NSF grant DMS-0436341 and the research of Yang Kuang is partially sup-ported by DMS-0436341 and DMS/NIGMS-0342388. We would like to thank William Fagan for initiatingthis interesting modeling project and the NCEAS grant that partially supported this work. We are alsovery grateful to Akiko Satake and Andrew Liebhold for many helpful discussions.

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Nature, London, 225, 58-60.[2] D. M. Johnson, A. M. Liebhold, P. C. Tobin and O. N. Bjornstad, 2006. Allee effects and pulsed invasion by the gypsy moth. Nature,

444, 361-363.[3] J. Guckenheimer, 1979. Sensitive dependence on initial conditions for unimodal maps. Communications in Mathematical Physics,70,

133-160,[4] L. Edelstein-Keshet, 2005. Mathematical models in biology, SIAM, Philadelphia.[5] R. M. May, 1973. On the relationship between various types of population models. American Naturalist, 107, 46-57.[6] R. M. May and G. F. Oster, 1976. Bifurcations and dynamic complexity in simple ecological models. The American Naturalist, 110,

573-599.[7] J. Maynard Smith and M. Slatkin, 1973. The stability of predator-prey systems. Ecology, 54, 384-391.[8] A. J. Nicholson, 1933. The balance of animal populations. Journal of Animal Ecology, 2, 131-178.[9] A. J. Nicholson and V.A. Bailey, 1935. The balance of animal populations. part I. Proceedings of the Zoological Society of London,

3, 551-598.[10] W. E. Ricker, 1954. Stock and recruitment. Journal of the Fisheries Research Board of Canada, 11, 559-623.[11] S. Wiggins. 2002. Introduction to applied nonlinear dynamical systems and chaos. 2nd Edition, Springer.[12] http://www.mda.state.md.us/plants-pests/forest_pest_mgmt/gypsy_moth/current_conditions.php

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