Numerical Bifurcation And Stability Analysis Of An Predator-prey System With Generalized Holling Type III Functional Response

abstract: We perform a bifurcation analysis of a predator-prey model with Holling functional response. The analysis is carried out both analytically and numerically. We use dynamical toolbox MATCONT to perform numerical bifurcation analysis. Our bifurcation analysis of the model indicates that it exhibits numerous types of bifurcation phenomena, including fold, subcritical Hopf, cusp, BogdanovTakens. By starting from a Hopf bifurcation point, we approximate limit cycles which are obtained, step by step, using numerical continuation method and compute orbitally asymptotically stable periodic orbits.


Introduction
The dynamical relationship between predators and their preys is one of the dominant subjects in ecology and mathematical ecology due to its universal importance, see [3].Mathematical modelling is an important and very useful tool in this field to get insight the dynamical behaviour of predator-prey systems.The key element in predator-prey interaction is "predator functional response on prey population", which is a function describing the number of prey consumed per predator per unit time for given quantities of prey and predator population.When there is no predator, the logistic equation models the behavior of the preys.For interactions between preys and predators, we use the generalized Holling response function of type III.These models with Holling functional response are further 90 Z. Lajmiri, R. K. Ghaziani, M. Ghasemi studied by many authors, see [10], [14], [15].This response function which models the consumption of preys by predators is such that the predation rate of predators increases when the preys are few and decreases when they reach their satiety.The main aim of this paper is to study the pattern of bifurcation that takes place as we vary some of the model parameters.We specially focus on the biological implications of the found bifurcations.Most importantly we show that the Hopf bifurcation plays, for various reasons, a crucial role.Ecological systems are complex because of the diversity of biological species as well as the complex nature of their interactions.In this work we will use the term complexity to describe the ecological complexity found in nature as well as the of dynamical complexity of the models.Although, only chaotic dynamics are called complex, we can say that periodic behavior is more complex than stationary behavior.Quasiperiodic behavior is more complex than periodic behavior but is less complex than chaotic behavior.It is interesting to note that the instability of steady-state for temporal models of predator-prey interaction leads to either oscillatory coexistence state or extinction of one or both the species [4], [6], [16].This is actually done by studying the change in the eigenvalue of the Jacobian matrix and also following the continuation algorithm.Numerical bifurcation analysis techniques are very powerful and efficient in physics, biology, engineering, and economics.The characteristics of Hopf point, the limit cycle and the general bifurcation may be explored using the software package MAT-CONT [5].This package is a collection of numerical algorithms [1], implemented as a MATLAB toolbox for the detection, continuation and identification of limit cycles.In this package we use prediction-correction continuation algorithm based on the Moore-Penrose matrix pseudo inverse for computing the curves of equilibria, limit point (LP), along with fold bifurcation points of limit point (LP) and continuation of Hopf point (H), etc.In thise paper we rely heavily on advanced numerical methods, namely numerical continuation to obtain results that cannot be obtained analytically.
The paper is organized as follows: In Section 2, we introduce the model and discuss the stability and bifurcations of its equilibrium points.In Section 3, we numerically compute curves of codim-1 bifurcations and the critical normal form coefficients of codim-2 bifurcation points, using the MATLAB toolbox MTATCONT.In Section 4, we summarize our results.

The mathematical model and its stability analysis
Recently, the Leslie predator-prey model has received some interest, see [2], [7], [9], [12], [18].We take the general form of Leslie predator-prey model as ) , x(0) > 0, y(0) > 0. ( ).Then the sysyem takes the following form: ( where the parameters r, K, m, a, s and h are all positive constants, and b is negative and b > −2 √ a.Before going into details, we make the following rescaling: ). (2.3) This system first was studied in [11], in which some local bifurcations were studied analytically.
In this section, we analyze the local stability for the positive equilibrium 3).The Jacobian matrix of system (2.3) at E 1 : The trace of J E1 is If condition tr(J E1 ) < 0 holds, then the positive equilibrium E 1 of system (2.3) is a locally asymptotically stable node or focus; if condition tr(J E1 ) > 0 holds, and the positive equilibrium E 1 of system (2.3) is an unstable node or a focus.Hence, we have the following results.
Lemma 2.3.Let us assume the following mutually exclusive conditions hold: then the positive equilibrium E 1 is an unstable focus Figs.5,6.

Continuation Curve of Equilibrium Points
The main aim of this section is to study the pattern of bifurcation that takes place as we vary the parameters δ, a and β.This is actually done by studying the change in the eigenvalue of the Jacobian matrix and also following the continuation algorithm.To start with, we consider a set of fixed point initial solution, (x, y) = (0.39, 0.58), corresponding to a parameter set of values, a = 10, b = −5.5, β = 0.333333, δ = 0.5.The characteristics of Hopf point, the limit cycle and the general bifurcation may be explored.To compute curve of equilibrium from the equilibrium point we take δ as the free parameter with fixed a = 10, b = −5.5 and β = 0.333333.From Fig. 1, 2, it is evident that the system has a Hopf point (H) and two limit point (LP) at: label = LP, x = ( 0.499996 0.749994 0.499999 ) a=1.870593e+004 label = H , x = ( 0.499999 0.749999 0.499999 ) Neutral saddle label = LP, x = ( 0.430074 0.641692 0.497349 ) a=-3.746749e+000 label = H , x = ( 0.400001 0.600001 0.499999 ) First Lyapunov coefficient = 6.765879e+000

Concluding remarks
In this paper, we studied a planar system that models a predator-prey interaction with generalized Holling type III functional response.We derived analytically a complete description of the stability regions of the equilibrium points of the system, namely, E 3 , and E 1 .We showed that the system undergoes fold, Hopf, cusp and Bogdanov-Takens bifurcations.To support the analytical results and reveal the further complex behaviours of the system, we employed numerical continuation methods to compute curves of codimension 1 and 2 bifurcation points.We have shown that the ustable point equilibrium of the system undergoes a subcritical hopf bifurcation and becomes stable, while stable population cycles emerge.
2 then the following statements hold: (a) If δ < δ * , the positive equilibrium E 3 is locally asymptotically stable.(b) If δ > δ * , the positive equilibrium E 3 is unstable.(c) If δ = δ * , then a Hopf bifurcation occurs around the positive equilibrium E 3 .

Figure 3 :
Figure 3: (a) Limit cycles starting from the Hopf point, (b) Limit cycles and bifurcations of limit cycles.

Table 2 :
One-parameter bifurcation points and eigenvalues.

Table 3 :
Bifurcations of limit cycles and eigenvalues, the family of limit cycles bifurcating from the Hopf point.