Analysis of the dynamics of a realistic ecological model

Christophe Letellier

2002, Chaos, Solitons & Fractals

Abstract

A fairly realistic three-species food chain model based on the Leslie±Gower scheme is investigated by using tools borrowed from the nonlinear dynamical systems theory. It is observed that two co-existing attractors may be generated by this ecological model. A type-I intermittency is characterized and a homoclinic orbit is found. Ó : S 0 9 6 0 -0 7 7 9 ( 0 0 ) 0 0 2 3 9 -3

Figures (15)

where a, is the rate of the self-growth for prey x; a measures the rate at which y will die out when no x remain; «w;’s are the maximum value which per capita rate can attain; dy) and d, signify the extent to which environment provides protection to the prey x; b; measures the strength of competition among individuals of the species x; d) is the value of y at which per capita removal rate of » becomes w,/2; d; represents the residual loss in Z population due to severe scarcity of its favorite food y; co describes the growth rate of the generalist predator Z by sexual reproduction, the number of males and females being assumed to be =qual. A quite realistic model involving a three-species food chain is considered. A prey species x serves as the only food for the predator ~ which is itself predated by another predator z. A typical situation of such a scheme would involve rodents, snakes and peacocks [11]. Interaction between the specialist predator » and its prey ¥ may be modeled by the Volterra scheme, i.e., predator population dies out exponentially in the absence of its prey. Nevertheless, for a more realistic model, the interaction between this predator y and the generalist predator Z is rather modeled by the Leslie-Gower scheme where the loss in a predator population is proportional to the reciprocal of per capita availability of its most favorite food. A set of ordinary differential equations governing the above model is then given by [12]:

The parameter values are chosen on the basis of previous studies [12] and correspond to quantitative measures of attributes of the rodent-snake—peacock food chain. The a -bifurcation parameter will be varied during the present analysis. The system (3) has six fixed points. One of them, Fo, is located at the origin of the phase space R*(x,y,z). The others are: These six fixed points are located on the xy-plane projection for a, = 1.93 with a chaotic trajectory (Fig. 1). Fixed points Fo and F;, always exist. It may be shown that F; never exists in the positive octant [7]. The existence of the fixed points Fy. have been studied by investigating their local and global stability [7]. Since, for the parameters given in (5), it is located in the positive octant, the fixed point F3 will be designated as the ‘inner fixed point’.

Fig. 1. Plane projection of the chaotic attractor observed for a; = 1.93. Three fixed points are also reported. F; is located very far from the attractor and not in the positive octant. The fixed points Fy, are imaginary. None of the latter play any role in the dynamics.

Fig. 2. The bifurcation diagram versus the a,-control parameter of the ecological model for c3 = 0.03. As usual, an intermittent behavior is observed before the periodic window. It results from a tangent bifurcation associated with the saddle-node bifurcation. In order to avoid numerical difficulties in computing the Poincaré section, the first-return map (Fig. 3(c)) is computed in the Poincaré section, defined as

Fig. 3. First-return maps to the Poincaré section for different values of the bifurcation parameter a}. A unimodal map with a dif- ferentiable maximum is found after the accumulation point as predicted by the kneading theory. For large a,-values, a bimodal map is observed up to a tangent bifurcation for a, = 1.977403. (a) a; = 1.8660; (b) a; = 1.93; (c) a; = 1.977403.

Fig. 4. Intermittent behavior is observed for a; = 1.97740. Laminar phases during which the behavior is almost periodic are inter- rupted by chaotic bursts (co = 0.03).

Fig. 5. Distribution of the laminar length / for a; = 1.97740 and c3 = 0.03. The histogram is characteristic of a type-I intermittency.

Fig. 6. Mechanisms responsible for the destruction of the periodic orbits. The periodic orbits are created by two saddle-node bifur- cations through the critical point C;. In the case of the Rossler system, the orbit (20011) is destroyed with the orbit (10011) and (20010) with (10010). This feature comes from the fact that the destruction is governed by the critical point C). Contrary to this, in the case of the ecological model, the critical point C, has a very limited role and the destruction of the orbits is induced by the critical point C). Consequently, the orbit (10011) is destroyed with the orbit (10010), and (20011) with (20010). (a) Rossler system. (b) Ecological model. Population of periodic orbits embedded within the attractor for two values of the a; bifurcation parameter (co = 0.03)

Fig. 7. First-return maps to the Poincaré section P, for three different high bifurcation parameter values. (a) a; = 2.10; (b) ay = 2.20; (c) a; = 3.02. —e EE ee ee ee eee When a; €]2.10; 3.02], it is rather difficult to compute a safe Poincaré section, i.e., without any spu- rious intersections between the trajectory and the Poincaré plane. These difficulties arise from the fact that there is no plane projection where a hole may be found in the middle of the attractor. As a con- sequence, spurious intersections cannot be avoided. For instance, for a; = 2.20 (Fig. 7(b)), a piece of ¢ monotonic branch is observed near the bissecting line and located at the right side of the first-return map Such a small branch is supicious. For these reasons, it may be difficult to describe the pruning process fo1 periodic orbits. Nevertheless, up to five monotonic branches could be possibly identified on this range o! ai-values.

respectively. For instance, for a; = 0.17924, a period-1 limit cycle coexists with a period-8 limit cycle (Fig. 10). Fig. 8. When the growth-rate of the generalist predator co is increased, the saddle-node bifurcation creating the stable limit cycle (3) and the unstable one (2) occurs for smaller values of the self-growth rate of the prey x. The period-doubling bifurcation inducing the period-8 limit cycle encoded by (10111010) is slightly dependent on co.

Fig. 9. Bifurcations diagram of the three-species food chain model for co = 0.038. Two different attractors are observed when a, € {1.7804; 1.8915]. Two co-existing period-doubling cascades are then observed. The chaotic attractor issued from the main one is destroyed by a boundary crisis with the unstable periodic orbit (2) created by the saddle-node bifurcation observed for a, = 1.7804. The stable limit cycle (3) is destabilized through a period-doubling bifurcation followed by a period-doubling cascade for a; = 2.002.

Fig. 10. The period-1 limit cycle (3) coexists with the period-8 limit (10111010) cycle for a; = 1.7924. The small loop presented by the cycle (3) results from a projection effect and the limit cycle has rather a period equal to | as observed on the bifurcation diagram. For instance, another plane projection induced by the z-time series clearly confirms the periodicity of the limit cycle (3). This plane pro- jection also exhibits that the stable limit cycle has a period equal to 8 rather than 4 as suggested by the xy-plane projection. They are involved in two different period-doubling cascades which are concomitant (co = 0.038). (a) xy-plane projection. (b) X Y-plane pro- jection.

Fig. 11. Two co-existing limit cycles for c3 = 0.03. a. > ae Such a feature is particularly important since very often, as observed on the Rossler system [10], when he increasing branch reaches the bisecting line, a homoclinic orbit may be observed. A homoclinic orbit visits the small neighborhood of an inner fixed point, spaces out by its unstable manifold and is reinjected in he neighborhood of the same fixed point by its stable manifold. An existence criterion has been proposed by Sil’nikov [27]. A homoclinic orbit has been observed for co = 0.029101 and a, = 1.917420 (Fig. 12(a)). It is checked that the increasing branch of the unimodal first-return map then reaches the bissecting line (Fig. 2(b)) as expected. In the present case, the trajectory visits a neighborhood where the derivatives become very small but, surprisingly, no fixed point has been identified in this neighborhood. For instance, the rajectory visits the point P which has the coordinates

Fig. 12. An approximation of the homoclinic orbit observed on the three-species food chain model for a; = 1.917420 and co = 0.029101. The trajectory visits the neighborhood of the inner fixed point /4,. The first-return map exhibits an increasing branch which reaches the bisecting line. (a) XY-plane projection. (b) First-return map.

Fig. 13. Time evolution of the different normalized species in a homoclinic situation (a; = 1.917420 and co = 0.029101).