Received: 27 March 2019 Revised: 24 October 2019 Accepted: 4 December 2019
DOI: 10.1002/cmm4.1079
RESEARCH ARTICLE
Global dynamics of a tritrophic food chain model subject to
the Allee effects in the prey population with sexually
reproductive generalized-type top predator
Surajit Debnath Uttam Ghosh Susmita Sarkar
Department of Applied Mathematics,
University of Calcutta, Kolkata, India
Correspondence
Uttam Ghosh, Department of Applied
Mathematics, University of Calcutta,
Kolkata-700 009, India.
Email: uttam_math@yahoo.co.in
In this article, we have considered a continuous tritrophic food chain model
subject to the Allee effect on the prey growth function with prey-dependent
Holling type II functional response between the prey and intermediate preda-
tor; Crowley-Martin senses functional response between intermediate predator
and top predator, and the top predator is of sexually reproductive type. We
have established the positivity and boundedness of the system and the condi-
tion of existence of different equilibrium points. The local and global stability
of the solutions about the various equilibrium points has been investigated. The
center manifold theorem has been used to find the nature of the solution in
the neighborhood of the nonhyperbolic equilibrium points and the direction of
Hopf bifurcations. Numerical simulation has been carried out to establish the
theoretical findings and finally some concluding remarks are given.
KEYWORDS
Allee effect, center manifold theorem, chaotic behavior, Crowley-Martin senses functional response,
global stability, Holling type II functional response, Hopf bifurcation, saddle-node bifurcation
1 INTRODUCTION
Development of ecological systems has become an increasingly active research area of ecological science. Predator-prey
interaction is one of the basic issues in ecological and social science. A lots of pioneer work has been done on the
Lotka-Volterra predator prey model.
1-4
One of the most valuable features in predator-prey interaction is the functional
response. The functional response of the predator to the prey density is the function that describes the amount of prey
consumed by a predator per unit time. From earlier, various forms of functional responses have been applied to formulate
different ecological models. In the classical Lotka-Volterra model,
5,6
it was assumed that the functional response is directly
proportional to the product of prey and predator density. This model has fundamental importance for the predator-prey
interaction. Then, Holling suggests three important types of functional response (Holling types I, II, III). One of the great
significant prey-dependent functional response is the Holling type II functional response. Here, the per-capita consump-
tion of the predator to the prey is a function of the prey density only. A few numbers of practical experiment have strongly
supported the prey-dependent functional response.
7-9
This type of functional response has a wide field of implementation
in the ecological science.
10-12
There is another important type of functional response called the Crowley-Martin senses functional response.
13
A sev-
eral laboratory and field experiments suggest that when the number of predators is large and the competition among
the predators affects the predation, then the Crowley-Martin senses functional response suggests the better results in the
Comp and Math Methods. 2020;2:e1079. wileyonlinelibrary.com/journal/cmm4 © 2019 John Wiley & Sons, Ltd. 1 of 23
https://doi.org/10.1002/cmm4.1079