Coexistence in competition models with density-dependent mortality (original) (raw)

Global dynamics of a predator–prey model

Journal of Mathematical Analysis and Applications, 2010

This paper deals with the dynamics of a predator-prey model with Hassell-Varley-Holling functional response. First, we show that the predator coexists with prey if and only if predator's growth ability is greater than its death rate. Second, using a blow-up technique, we prove that the origin equilibrium point is repelling and extinction of both predator and prey populations is impossible. Third, the local and global stability of the positive steady state coincide when the predator interference is large. Finally, for a typical biological case, we show instability of the positive equilibrium implies global stability of the limit cycle. Numerical simulations are carried out for a hypothetical set of parameter values to substantiate our analytical findings.

A Mathematical Model to Study Stability of Biological Interaction: Competition

International Journal of Mathematics Trends and Technology, 2017

In this paper we shall construct a mathematical model to study the impact of competition on the growth of two species in a given geographical region and for a given interval of time. In our paper we shall also discuss the stability of autonomous dynamical system representing the growth of two species and study the relation between the population density of competitively superior species and competitively inferior species competing for the same resources in a given region and in a given interval of time

Stability of a one-predator two-prey system governed by nonautonomous differential equations

arXiv: Dynamical Systems, 2015

A non-periodic version of the one-predator two-prey system model presented in [L.T.H. Nguyen, Q.H. Ta, T.V. T\d{a}, Existence and stability of periodic solutions of a Lotka-Volterra system, SICE International Symposium on Control Systems, Tokyo, Japan, 712-4 (2015) 1-6] is considered. First, we prove existence of unique positive solutions to the model. Second, we show existence of an invariant set, which suggests the survival of all species in the system. On the other hand, we show that when the densities of two prey species are quite small, the predator falls into decay. Third, we explore global asymptotic stability of the system by using the Lyapunov function method. Finally, some numerical examples are given to illustrate our results.

Conditions for permanence in well-known biological competition models

The ANZIAM Journal, 2000

Reaction-diffusion systems are widely used to model the population densities of biological species competing for natural resources in their common habitat. It is often not too difficult to establish positive uniform upper bounds on solution components of such systems, but the task of establishing strictly positive uniform lower bounds (when they exist) can be quite troublesome. Two previously established criteria for the permanence (non-extinction and non-explosion) of solutions of general weakly-coupled competition-diffusion systems with diagonally convex reaction terms are used here as background to develop more easily verifiable and concrete conditions for permanence in various well-known competitiondiffusion models. These models include multi-component reaction-diffusion systems with (i) the by now classical Lotka-Volterra (logistic) reaction terms, (ii) higher order "logistic" interaction between the species, (iii) logistic-logarithmic reaction terms, (iv) Ayala-Gilpin-Ehrenfeld ^-interaction terms (which are used to model Drosophila competition), (v) logistic-exponential interaction between the species, (vi) Schoener-exploitation and (vii) modified Schoener-interference between the species. In (i) a known condition for permanence (for the ODE-system) is recovered, while in (ii)-(vii) new criteria for permanence are established.

A Stability Analysis on Models of Cooperative and Competitive Species

This paper presents a stability analysis on generalised mathematical models for cooperative and competitive species. For each system, we determined all the relevant equilibrium points and analysed the behavior of solutions whose initial conditions satisfy either x1= 0 or x2= 0. The curves in the phase plane along which the vector field is either horizontal or vertical were determined. For each of the systems, we described all possible population scenarios using the phase potraits. The cooperative system was found to be stable at one of the two equilibrium points presents and unstable (Saddle) at the other. Four equilibrium points existed for the competitive species model for which the system is stable at one point and locally asymptotically stable at the other three points. The asymptotical stability is based on the inhibition and the coexistence factors between the two competing species.

Effect of the mortality on a density-dependent model with a predator-prey relationship

2020

In this work, we consider an inter-specific density-dependent model of two species competing on a single nutrient in a chemostat, taking into account the prey-predator relationship. With different dilution rates, we give a complete analysis of the existence and local stability of all the steady states. Indeed, under the joint effect of competition with the prey-predator relationship and mortality, we show that the positive steady state can be destabilized by a supercritical Hopf bifurcation with the appearance of a stable limit cycle. However, without mortality, there is a stable persistence of the coexistence steady state where there cannot be periodic orbits.

Global dynamics of a predator, weaker prey and stronger prey system

Applied Mathematics and Computation, 2015

In this paper, we propose and analyze a prey-predator system consisting of two competitive prey populations and one predator population which depends on both the prey species. We investigate the boundedness and persistence criteria of the system and existence conditions of all the possible equilibria. Further the dynamical behavior from the point of view of local and global stability at different equilibria are presented. We also determine the explicit conditions so that the system has no periodic solutions. Finally, we present some numerical examples to illustrate our analytical works.