Dynamical complexities and pattern formation in an eco-epidemiological model with prey infection and harvesting
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Dynamical complexities and pattern formation in an eco-epidemiological model with prey infection and harvesting S. N. Raw1
· Barkha Tiwari1 · P. Mishra2
Received: 9 December 2019 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this work, we have studied the emergence of complex dynamics in a hypertrophic lake ecosystem. We have proposed and analysed a three-tier spatio-temporal predator– prey model in which the prey population is influenced by infection, diffusion, and harvesting. In the absence of diffusion, existence conditions of biologically feasible equilibrium points with their stability properties and bifurcations have been discussed. We observe that the non-diffusive system exhibits a variety of bifurcations such as saddle-node, transcritical and undergoes for a supercritical Hopf-bifurcation. In the spatio-temporal system, all possible conditions for Turing instability are derived. The complex behaviour of the system is shown through some interesting numerical examples. Phase portraits, time evolution diagrams, and numerical bifurcation diagrams show the ecological complexity of the degraded lake ecosystem. Diffusion driven spatio-temporal complexity is investigated with the help of pattern formulation. Various spatial patterns reveal that degradation has the potential to alter fish spatial distributions in the lake. Keywords Complex dynamics · Pattern formulation · Hopf-bifurcation · Turing instability · Degraded ecosystem Mathematics Subject Classification 92D25 · 92D30 · 92D40 · 34D23 · 37G15
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S. N. Raw [email protected] Barkha Tiwari [email protected] P. Mishra [email protected]
1
Department of Mathematics, National Institute of Technology, Raipur, Chhattisgarh 492010, India
2
Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
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1 Introduction In the last few decades, mathematical ecology has made its own identity in the world of science and technology, which is gaining the attention of researchers. Ecoepidemiology is the branch of mathematical ecology, which is a new cross-field of ecology and epidemiology. Mathematical models are a powerful tool for investigating the long term behaviour of interacting populations in the ecosystem. The fascinating part of the study is to understand the processes of the ecosystem which provides a better means to balance the level of biodiversity. After the pioneering work of Lotka and Volterra on the population dynamics, Kermack and McKendrick [1] proposed a new idea to study the behaviour of the population influenced by the disease through a classical susceptible, infectious, recovered model (SIR model). Anderson and May [2] studied an eco-epidemiological model of the micro-parasitic disease caused by viruses or bacteria and helminths or arthropods, respectively. There is huge literature available when we look into predator–prey models with disease and infection. Packer et al. [3] proposed and studied the model for micro-para
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