Role of time delay and harvesting in some predator–prey communities with different functional responses and intra-species competition

ABSTRACT

We propose four predator–prey models: RM (Rosenzweig–MacArthur) model, BD model (RM type model with Beddington–DeAngelis functional response), RMI model (i.e., RM model with intraspecific competition among predators) and BDI model (BD model with intraspecific competition among predators). Each model incorporates time delay in the predators’ numerical response. We first analyse the delay-induced stability for all the models. We show that increasing delay always destabilizes a coexisting stable equilibrium in RM and BD models. However, increasing delay does not always destabilize a stable equilibrium in RMI and BDI models. Indeed, the stable equilibrium, in the latter two models, may also maintain its stability due to varying delay. Thus, one of the major conclusions is that the invariance property of the local stability in RMI and BDI models is due to the influence of intraspecific competition. Analytically, we prove that stability switching is impossible to occur in all the models. Later, we implement harvesting of the prey and predator separately, which may generate stability switching. If populations oscillate in the unharvested system, extensive effort has a potential to stabilize the equilibrium. Under the same natural condition (unharvested situation), prey harvesting and predator harvesting may produce opposite dynamic modes.

KEYWORDS:

• Delay differential equation
• Beddington-DeAngelis functional response
• Hopfbifurcation
• harvesting

Acknowledgments

B.B. expresses gratitude to the MHRD, Government of India, for its financial assistance to pursue her PhD. B.G. acknowledges the financial support received from SERB, Government of India, under Core Research Grant (Ref. No. CRG/2020/005621). We are thankful to the esteemed reviewers, and Prof. Jie Shen, the Editor-in-Chief, for their valuable comments and suggestions, which helped in improving the work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by (i) the Ministry of Human Resource Development (MHRD) and (ii) Science and Engineering Research Board (SERB) [File No.: CRG/2020/005621].

Notes on contributors

Binandita Barman

Ms. Binandita Barman is working as a research scholar in the Department of Mathematics,National Institute of Technology Meghalaya, India. She did her M.Sc from Tezpur University, India in 2015. Her research interest includes Delay Differential Equations and Theoretical Ecology. She has four articles published in Chaos, Solitons, and Fractals (Elsevier), Ecological Complexity (Elsevier), and International Journal of Modeling and Simulation (Taylor & Francis).

Bapan Ghosh

Dr. Ghosh is an Assistant Professor of Mathematics atIndian Institute of Technology Indore, India. Earlier he had worked as an Assistant Professor at the Department of Mathematics, National Institute of Technology Meghalaya, India. He completed his MSc in Applied Mathematics from The University of Burdwan, and PhD in Nonlinear Dynamics and Mathematical Biology from Indian Institute of Engineering Science and Technology, Shibpur, India. He worked as a short- and long-term researcher in France, Taiwan, and Russia. His area of research includes Mathematical Biology, Dynamical Systems, Delay and Fractional Differential Equations, Numerical Analysis, etc. He has published several articles in international journal of repute. He is guiding PhD scholars and postdoctoral researchers. He is also involved in several national and international projects.