Fisher-KPP dynamics in diffusive Rosenzweig–MacArthur and Holling–Tanner models

Abstract

We prove the existence of traveling fronts in diffusive Rosenzweig-MacArthur and Holling-Tanner population models and investigate their relation with fronts in a scalar Fisher-KPP equation. More precisely, we prove the existence of fronts in a Rosenzweig-MacArthur predatorprey model in two situations: when the prey diffuses at the rate much smaller than that of the predator and when both the predator and the prey diffuse very slowly. Both situations are captured as singular perturbations of the associated limiting systems. In the first situation we demonstrate clear relations of the fronts with the fronts in a scalar Fisher-KPP equation. Indeed, we show that the underlying dynamical system in a singular limit is reduced to a scalar Fisher-KPP equation and the fronts supported by the full system are small perturbations of the Fisher-KPP fronts. We obtain a similar result for a diffusive Holling-Tanner population model. In the second situation for the Rosenzweig-MacArthur model we prove the existence of the fronts but without observing a direct relation with Fisher-KPP equation. The analysis suggests that, in a variety of reaction-diffusion systems that rise in population modeling, parameter regimes may be found when the dynamics of the system is inherited from the scalar Fisher-KPP equation.Reaction-diffusion systems are often used in population dynamics modeling when it is desirable to take into account random motion of individuals in the population. In systems where there is more than one spatially homogeneous equilibrium state, it is of interest to know whether transition fronts between these states exist. We present analysis of traveling fronts in two diffusive populationdynamics models, one of which is a modified Rosenzweig-MacArthur model, and the other is Holling-Tanner predator-prey model. The analysis is performed in detail on the Rosenzweig-MacArthur system (Section 2), while for the Holling-Tanner system (Section 6) most of the details are skipped and similarities in the proofs are pointed out. The plan of the paper is as follows. We introduce the Rosenzweig-MacArthur model in Section 2.1 and explain the results of the paper and their mathematical implications. The scaling of the model that we use and the parameter regimes that the analysis covers are described in Section 2.2. The regimes are grouped in two cases that are then analyzed using geometric singular perturbation theory in Sections 3 and 4. In Section 3 the relation of the fronts with a scalar Fisher-KPP equation is revealed. In the analysis of the Rosenzweig-MacArthur model, assumptions are made about some of the parameters, in order to simplify some of the calculations. In Section 5, we describe the implications of the assumptions and Date

More traveling waves in the Holling-Tanner model with weak diffusion

“….6) On the invariant line r = 0, (3.6) has the unique equilibrium E 2 = (0, 0) with eigenvalues 0 and −αβδ. A center manifold is the x 1 -axis, on which the system reduces to ẋ1 = x 2 1 . Combining the results from the two coordinate systems, we obtain Figure 3.3, which shows the flow near the degenerate equilibrium (0, 0) of (3.4) coordinates.…”

“…Turing bifurcations of the PDE (1.1) are studied in [14]. Existence of traveling front solutions that connect constant states of the PDE (1.1) has been established in [1,2,7], and existence of a family of wave train solutions was established in [7]. The importance of wave train solutions in ecological models is emphasized in [21].…”

Dynamics in diffusive Leslie–Gower prey–predator model with weak diffusion

“…Under our assumption 0 < δ 1, it is clear that system ( 6) is a multiscale slowfast system. Thus, using the method in [5,13,24], we reduce the multiscale slow-fast system (6) twice by the geometric singular perturbation. The first reduction is respect to smaller parameter , and the second is respect to δ.…”

“…For high dimension model, Liu, Xiao and Yi [23] considered the relaxation oscillation cycle of a slow-fast prey-predator model with one prey and two competing predators, and Shen, Hsu and Yang [27] studied the dynamics of a slow-fast intraguild predation model. For reaction-diffusion cases, Ducrot, Liu and Magal [11] studied the large speed traveling waves for the diffusive Rosenzweig-MacArthur predator-prey model, and Cai, Ghazaryan and Manukian [5] studied the travelling waves for the diffusive Rosenzweig-MacArthur and Holling-Tanner models with two small parameters. Furthermore, the geometric singular perturbation theory is an useful analysis method in these paper.…”