Global Stability Of Coexistence Equilibrium Research Articles

Predator-prey systems with defense switching and density-suppressed dispersal strategy.

In this paper, we consider the following predator-prey system with defense switching mechanism and density-suppressed dispersal strategy $ \begin{equation*} \begin{cases} u_t = \Delta(d_1(w)u)+\frac{\beta_1 uvw}{u+v}-\alpha_1 u, & x\in \Omega, \; \; t>0, \\ v_t = \Delta(d_2(w)v)+\frac{\beta_2 uvw}{u+v}-\alpha_2 v, & x\in \Omega, \; \; t>0, \\ w_t = \Delta w-\frac{\beta_3 uvw}{u+v}+\sigma w\left(1-\frac{w}{K}\right), & x\in \Omega, \; \; t>0, \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0, & x\in\partial\Omega, \; \; t>0, \\ (u, v, w)(x, 0) = (u_0, v_0, w_0)(x), & x\in\Omega, \ \end{cases} \end{equation*} $ where $ \Omega\subset{\mathbb{R}}^2 $ is a bounded domain with smooth boundary. Based on the method of energy estimates and Moser iteration, we establish the existence of global classical solutions with uniform-in-time boundedness. We further prove the global stability of co-existence equilibrium by using the Lyapunov functionals and LaSalle's invariant principle. Finally we conduct linear stability analysis and perform numerical simulations to illustrate that the density-suppressed dispersal may trigger the pattern formation.

Memory effect on Bazykin’s prey-predator model: Stability and bifurcation analysis

This paper deals with a system of two fractional order differential equations for prey-predator interaction with intra-specific competition among predators. The fractional order differential equation is considered in the sense of Caputo derivative and the derivation of the fractional order model is explained in terms of memory effect on population growth. Detailed mathematical results are provided to establish the positiveness, existence–uniqueness and boundedness of the solutions. The conditions required for local asymptotic stability of various equilibrium points and global stability of coexistence equilibrium are derived along with the Hopf-bifurcation condition for coexistence equilibrium. The effect of memory on the system dynamics through the shift of Hopf-bifurcation threshold is demonstrated with the help of exhaustive numerical simulations. This study also reveals the effect of memory based growth on global bifurcation threshold.

A Host-Parasite System with Multiple Parasite Strains and Superinfection Revisited: The Global Dynamics.

In this paper, we revisit a host-parasite system with multiple parasite strains and superinfection proposed by Nowak and May (Proc R Soc Lond B 255(1342):81-89, 1994), and study its global dynamics when we relax the two strict conditions assumed therein. As for system with two parasite strains, we derive that the basic reproduction number [Formula: see text] is the threshold condition for parasite extinction and the invasion reproduction number [Formula: see text] is the subthreshold condition for coexistence of two parasite strains. As for system with three parasite strains, we are surprised to discover the global stability of parasite-free and coexistence equilibrium, which is distinct from the previous result. Furthermore, for system with n strains, we obtain the global asymptotical stability of the parasite-free equilibrium, conjecture a general result on the global stability of coexistence equilibrium and provide two numerical examples to testify our conjecture.

Analysis of a Prey-Predator Fishing Model for Harvesting with The Limited Effect of Density and Toxicity

We proposed a model of prey-predator fishing by considering proportion of prey density and toxicity in ecosystem. The model is analysed to study about biological equilibrium, bionomic equilibrium and its stability. The aim of the model is to determine the optimal sustainable harvesting for each species. The optimal harvesting is resulted from Pontryagin’s maximal principle. The global stability of coexistence equilibrium is analyzed from Lyapunov function. The effect of toxicity leads to the decreasing of sustainable harvesting.

Global stability of the coexistence equilibrium for a general class of models of facultative mutualism

ABSTRACTMany models of mutualism have been proposed and studied individually. In this paper, we develop a general class of models of facultative mutualism that covers many of such published models. Using mild assumptions on the growth and self-limiting functions, we establish necessary and sufficient conditions on the boundedness of model solutions and prove the global stability of a unique coexistence equilibrium whenever it exists. These results allow for a greater flexibility in the way each mutualist species can be modelled and avoid the need to analyse any single model of mutualism in isolation. Our generalization also allows each of the mutualists to be subject to a weak Allee effect. Moreover, we find that if one of the interacting species is subject to a strong Allee effect, then the mutualism can overcome it and cause a unique coexistence equilibrium to be globally stable.