87 research outputs found
Fisher-KPP dynamics in diffusive Rosenzweig-MacArthur and Holling-Tanner models
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 predator-prey 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
Bifurcation on diffusive Holling–Tanner predator–prey model with stoichiometric density dependence
This paper studies a diffusive Holling–Tanner predator–prey system with stoichiometric density dependence. The local stability of positive equilibrium, the existence of Hopf bifurcation and stability of bifurcating periodic solutions have been obtained in the absence of diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system, which are spatially homogeneous. In order to verify our theoretical results, some numerical simulations are carried out. 
Global stability in a diffusive Holling-Tanner predator-prey model
A diffusive Holling-Tanner predator-prey model with no-flux boundary condition is considered, and it is proved that the unique constant equilibrium is globally asymptotically stable under a new simpler parameter condition. (C) 2011 Elsevier Ltd. All rights reserved
A Holling-Tanner predator-prey model with strong Allee effect
We analyse a modified Holling-Tanner predator-prey model where the predation
functional response is of Holling type II and we incorporate a strong Allee
effect associated with the prey species production. The analysis complements
results of previous articles by Saez and Gonzalez-Olivares (SIAM J. Appl. Math.
59 1867-1878, 1999) and Arancibia-Ibarra and Gonzalez-Olivares (Proc. CMMSE
2015 130-141, 2015)discussing Holling-Tanner models which incorporate a weak
Allee effect. The extended model exhibits rich dynamics and we prove the
existence of separatrices in the phase plane separating basins of attraction
related to co-existence and extinction of the species. We also show the
existence of a homoclinic curve that degenerates to form a limit cycle and
discuss numerous potential bifurcations such as saddle-node, Hopf, and
Bogadonov-Takens bifurcations
Qualitative analysis on the diffusive Holling-Tanner predator-prey model
We consider the diffusive Holling–Tanner predator–prey model subject to the homogeneous Neumann boundary condition. We first apply Lyapunov function method to prove some global stability results of the unique positive constant steadystate. And then, we derive a non-existence result of positive non-constant steady-states by a novel approach that can also be applied to the classical Sel’kov model to obtain the non-existence of positive non-constant steady-states if 0 < p ≤ 1
Computational Study of Traveling Wave Solutions and Global Stability of Predator-Prey Models
In this thesis, we study two types of reaction-diffusion systems which have direct applications in understanding wide range of phenomena in chemical reaction, biological pattern formation and theoretical ecology. The first part of this thesis is on propagating traveling waves in a class of reaction-diffusion systems which model isothermal autocatalytic chemical reactions as well as microbial growth and competition in a flow reactor. In the context of isothermal autocatalytic systems, two different cases will be studied. The first is autocatalytic chemical reaction of order without decay. The second is chemical reaction of order with a decay of order , where and are positive integers and . A typical system is and involving three chemical species, a reactant A and an auto-catalyst B and C an inert chemical species. We use numerical computation to give more accurate estimates on minimum speed of traveling waves for autocatalytic reaction without decay, providing useful insight in the study of stability of traveling waves. For autocatalytic reaction of order with linear decay , which has a particular important role in biological pattern formation, it is shown numerically that there exist multiple traveling waves with 1, 2 and 3 peaks with certain choices of parameters. The second part of this thesis is on the global stability of diffusive predator-prey system of Leslie Type and Holling-Tanner Type in a bounded domain with no-flux boundary condition. By using a new approach, we establish much improved global asymptotic stability of a unique positive equilibrium solution. We also show the result can be extended to more general type of systems with heterogeneous environment and/or other kind of kinetic terms
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