Coexistence and harvesting optimal policy in three species food chain model with general Holling type functional response

Abstract

In this paper, we have discussed harvesting of prey and intermediate predator species. Both are subjected to Holling type I–V functional response. Conditions for local and global stability of the nonnegative equilibria are verified. The permanent coexistence criterion of the model system and existence of optimal equilibrium solution of the control problem are demonstrated. Maximum sustainable yield and maximal net present revenue are determined. To confirm analytical results, numerical solution has been carried out using the Matlab™ ODE solver ODE45 and the simulations show the model system reveals complex behavior (such as oscillations), which reflects the real situation. Recommendations for Resource Managers From our investigation of this study, we recommend to the management the following points. Coexistence of the three species with harvesting, or persistence of the model system is possible provided that good management(treatment) of some factors (such as harvesting rate, growth rate of species, etc.) are performed. The dynamics reveals complex behavior (such as oscillations), which reflects the real situation and it is sensitive to the above factors, especially the growth rate of the intermediate predator. The policy makers should recommend the optimal effort h * to be applied and the optimal stock ( x * , y * , z * ) to harvest. This indicates that maximum profit will attain while securing sustainability of the three species in the ecosystem.

Optimality and sustainability of delayed impulsive harvesting

“…By Lemma 3.1 the solutions to (1) satisfy (11). Linearizing the difference equation x n+1 = f (x n , x n−k ) around the positive equilibrium x * given in ( 14), we get (24). By Lemma 4.2, the yield is maximal whenever E = E opt = 1 − e −rT /2 , and the linearized difference equation becomes…”

“…The close connection of continuous models to difference equations has led to extensive study of discrete population models [22] including harvesting [23]. The fact that species do not naturally exist in isolation but coexist, compete or serve as prey for others, led to extensive literature on harvesting of a single or multiple populations in a food chain, and on optimal yields for exploited species [24,25,26,27]. Incorporating harvesting in systems of differential or difference equations includes the case of structured populations where selective harvesting is allowed [28,29].…”

“…If (25) holds, then so does (18), and the positive equilibrium exists. Reducing (1) to difference equation ( 11), we once again linearize (11) around x * and obtain (24), where by (25),…”

Numerical Analysis of Prey Refuge Effects on the Stability of Holling Type III Four-Species Predator-Prey System

“…Furthermore, if the solutions approach the critical point over time rather than just staying within a set radius, the critical point is asymptotically stable [7]. A lot of work has been done in establishing the stability of the equilibrium points of dynamical systems, both locally and globally [10]. Shireen [9], studied a four-species model with prey refuge for its dynamical behavior including the stabilities of its equilibrium points and carried out numerical simulations to confirm his analytical results.…”