Partial characterization of the global dynamic of a predator-prey model with non constant mortality rate

Abstract

In this paper we characterize partially the global dynamic of a predator prey model with non constant mortality rate. Concretely, we give necessary and sufficient conditions in order the system be dissipative and permanent. We study the global stability of the nontrivial equilibrium, when it is unique. We show that it is possible the existence of a unique periodic solution which arises from a supercritical Hopf bifurcation and end up through a subcritical Hopf bifurcation; suggesting that the model exhibits new dynamical features which are not present in the classical model; i.e., in the model with constant mortality rate.

The Rosenzweig–MacArthur Graphical Criterion for a Predator-Prey Model with Variable Mortality Rate

“…For m = 2.265, the cycle disappears through a supercritical PAH bifurcation and not subcritical, as claimed in [6].…”

“…With our parameter values, we find x 1 = 0.0390 and x 2 = 0.3905. Then, by solving the equation ψ(x i )) = h(x i ), i = 1, 2, with respect to m, we obtain the two same values of m given in [6]. Here again, one could write the particular form of the coefficient ρ given by (23) in the case of (45) and g, p, q given by ( 27).…”

“…Actually, for the CF model, as written for example in [6], α = δ − m, where δ is a constant, and thus depends on m. It is the case of the parameter values indicated in Line 2 of Table 7, for which the bifurcation diagram of Fig. 3 (a) was dressed in [6]. The authors gave there the bifurcation values m 1 = 0.5806 and m 2 = 2.265.…”

Dynamic Analysis of a Heterogeneous Diffusive Prey-Predator System in Time-Periodic Environment

“…Obviously, when c � δ, equation 3can be simplified to a constant death rate type. Prey-predator systems with this nonconstant death rate have been studied by some researchers [15][16][17]. Additionally, in order to understand patterns and the mechanisms of spatial distribution of interacting species, the dispersal process is taken into consideration [18][19][20].…”

Bifurcation analysis of a diffusive predator–prey system with nonconstant death rate and Holling III functional response

“…In [4], by making a bifurcation analysis of system (1.1) depending on all parameters, it was shown the existence of a homoclinic orbit which bifurcates generating a locally unique periodic orbit, which in turn belongs to the same connected component that contains the periodic orbit generated through an Andronov Hopf bifurcation of a nontrivial equilibrium. In [5], the authors make a comprehensive description of the global dynamics of (1.1). More detail results of system (1.1) one can refer to [6].…”