Abstract
A reaction–diffusion interacting species system with Beddington–DeAngelis functional response that has been proposed in the environment of mathematical ecology, which provides the rise to spatial pattern formation, is investigated and associated with the models of deterministic dynamics. The dynamical behaviour of a generalist predator–prey system with linear harvesting of each species and predator-dependent functional response is fully analyzed. Conditions of stability behaviour of the interior equilibrium point are established properly. Furthermore, we have recognized that the unique positive equilibrium point of the system is globally stable via appropriate Lyapunov function structure, which signifies that appropriate harvesting has no impact on the persistence property of the harvesting system. Also, we establish the conditions for the existence of bifurcation phenomena including a saddle-node bifurcation and a Hopf bifurcation. Subsequently, complete analysis regarding the impact of harvesting is carried out, and an interesting decision is that under some appropriate constraints, harvesting has immense impact on the final size of the interacting species. In addition, in accordance with Turing’s ideas on morphogenesis , our analysis shows that harvesting effort in a reaction–diffusion predator–prey system plays a vital function for geological conservation of interacting species. Finally, we discuss sufficient conditions for the existence of bionomic equilibrium point and the optimal harvesting policy attained by using the Pontryagin maximal principle.
Highlights
A reaction–diffusion interacting species system with Beddington–DeAngelis functional response that has been proposed in the environment of mathematical ecology, which provides the rise to spatial pattern formation, is investigated and associated with the models of deterministic dynamics. e dynamical behaviour of a generalist predator–prey system with linear harvesting of each species and predator-dependent functional response is fully analyzed
We have recognized that the unique positive equilibrium point of the system is globally stable via appropriate Lyapunov function structure, which signifies that appropriate harvesting has no impact on the persistence property of the harvesting system
In the course of stability theory of ordinary differential equations (ODEs), it has been verified that the interior equilibrium exists under certain conditions and it is globally asymptotically stable via suitable Lyapunov function structure, which signifies that appropriate harvesting has no impact on the persistence property of the harvesting system
Summary
Assuming the importance of harvesting on a generalist predator–prey model with Beddington–DeAngelis functional response, an effort is prepared to study the influence of linear harvesting of each species in a two-dimensional (2D) Beddington–DeAngelis predator–prey model which is a modification of the following model investigated extensively by Haque [24] and Sarwadi et al [25]. Proof (i) In order to prove that the system is locally asymptotically stable around the equilibrium point e3(u3, v3), firstly, we need to observe the Jacobian matrix J3 of the system around the point, which is given by. For better understanding of qualitative change of the temporal dynamics of the system (2), one may observe the bifurcation diagrams of both prey and predator species with respect to the system parameter ε (cf Figures 7 and 8). In support of this technique of bifurcation offered, the successive maxima of u and v in the ranges [0.01, 0.8] and [0.01, 0.33], respectively, as a function of ε in the range 0.95 ≤ ε ≤ 1.2 and the other system parameters are captioned in the figure. Due to the certain amount of prey harvesting from the interacting species system, predator species cannot consume prey species in earlier manner and this scenario supports the natural phenomena very well
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