Dynamics of a delayed predator‐prey model with Allee effect and Holling type II functional response

Abstract

In this paper, a delayed with Holling type II functional response (Beddington-DeAngelis) and Allee effect predator-prey model is considered.The growth of the prey is affected by the parameter M, which defines the Allee effect. In addition, the delay also influences the logistic growth of the prey, which can be interpreted as the maturity time or the gestation period. In the study of the characteristic equation, we observe that the delay also depends on the parameter M, which affects the dynamics in the prey population. Considering the delay as a bifurcation parameter, the local asymptotic stability of the positive equilibrium is investigated. On the other hand, we find that the system can also suffer a Hopf bifurcation in the positive equilibrium when the delay passes through a sequence of critical values. In particular, we study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions, an explicit algorithm is provided applying the normal form theory and center manifold reduction for the functional differential equations. Finally, numerical simulations that support the theoretical analysis are included. KEYWORDSAllee effect, delay predator-prey model, equilibrium point, Holling type II functional response, Hopf bifurcation, stability/unstablity MSC CLASSIFICATION 34K17; 34K18; 34K20Math Meth Appl Sci. 2020;43:5708-5728. wileyonlinelibrary.com/journal/mma

Discretization, Bifurcation, and Control for a Class of Predator-Prey Interactions

“…By contrast, a recent study led to the discrete dynamical system becoming more suitable than a continuous version when the population is non-overlapping (e.g., see, Jing et al [14], Liu et al [15], Lopez-Ruiz and Fournier-Prunaret [16], Neubert and Kot [17]). Furthermore, multiple existing studies related to the dynamics of predator-prey models are described in [18][19][20][21][22][23][24][25][26]. In [27], the Holling type-III functional response was introduced in both populations (prey and predator).…”

“…Furthermore, a detailed investigation of some charismatic population models and their qualitative behavior are provided (see, Din and Din et al [18][19][20][21][22][23][24][25][26] and the references therein).…”

Multiple dynamics in a delayed predator‐prey model with asymmetric functional and numerical responses

“…The combined impacts of both time delays and Allee effects are studied worldwide to understand the dynamics both the factors display. A delayed model of predator and prey (with Allee effect) has been investigated by Anacleto and Vidal 26 to show the existence of stability switching analytically and establish the direction of Hopf‐bifurcation with respect to time delay. They have found that the delay due to the gestation period can be effected by Allee effect, leading to a stability switching around the coexisting equilibrium.…”

“…They also determined that both the species become extinct due to larger Allee effect. On the other hand, in the predator‐prey model by Anacleto and Vidal, 26 the growth term of the prey is effected by both time delay due to maturation as well as by Allee effect. It was observed that Allee effect plays an important role in the existence of positive eqilibria.…”