Abstract
In this study, the dynamics of a diffusive Lotka–Volterra three-species system with delays were explored. By employing the Galerkin Method, which generates semi-analytical solutions, a partial differential equation system was approximated through mathematical modeling with delay differential equations. Steady-state curves and Hopf bifurcation maps were created and discussed in detail. The effects of the growth rate of prey and the mortality rate of the predator and top predator on the system’s stability were demonstrated. Increase in the growth rate of prey destabilised the system, whilst increase in the mortality rate of predator and top predator stabilised it. The increase in the growth rate of prey likely allowed the occurrence of chaotic solutions in the system. Additionally, the effects of hunting and maturation delays of the species were examined. Small delay responses stabilised the system, whilst great delays destabilised it. Moreover, the effects of the diffusion coefficients of the species were investigated. Alteration of the diffusion coefficients rendered the system permanent or extinct.
Highlights
The population dynamics of various biological and ecological systems have garnered much attention, promoting the emergence of mathematical models describing population dynamics and species interactions. These mathematical models must incorporate both spatial diffusion and time delays to mirror the dynamic nature of biological systems and the tendency of the species to move to the least densely populated areas
The delay-diffusive logistic equation describes the growth dynamics of a single species, whilst the delay-diffusive Lotka–Volterra predator-prey model describes the dynamics of multiple species [5,6,7]
Faria [10] investigated the predator-prey system with one or two time delays and a unique positive equilibrium. They examined the dynamics of this system based on the local stability of the equilibrium and the region of the Hopf bifurcation map that has been proven to occur when one of the delays is assumed to be a bifurcation parameter
Summary
The population dynamics of various biological and ecological systems have garnered much attention, promoting the emergence of mathematical models describing population dynamics and species interactions. Pao [22] studied a three species time-delayed Lotka–Volterra reaction–diffusion system and obtained certain conditions of the existence and global asymptotic stability of a positive steady-state solution. They showed that the three-species could coexist and that all trivial and semi-trivial solutions were unstable. Population dynamic issues are tied to both space and time, which implies that the state change will be impacted by both the present state and the past To this end, the present study aimed to analyse a three-species predator-prey Lotka–Volterra reaction–diffusion model by taking the effects of both diffusion and time delay into account.
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