Diffusion driven finite time blow-up and pattern formation in a mutualistic preys-sexually reproductive predator system: A comparative study

Abstract

Species have the general tendency to sustain their own survival chance in the ecology. In this paper, we design different diffusive model systems in which two prey populations make mutualistic relationship in which they are getting benefited from each other with their usual growth rate and sexually reproductive generalist predator preys upon the prey according to their functional responses. In the absence of generalist predator, there is no sign of mutualism between two prey. Sometimes environment may turn favourable for the invasive species, causing the growth of their population to outbreak. Biological control is an adopted strategy to limit harmful populations. To establish a control strategy that decreases the harmful population to healthy levels as opposed to high and risky levels, five different diffusive models have been introduced associated to their functional responses which are either prey dependent (Holling type-III and IV) or predator dependent (Beddington–DeAngelis, Crowley–Martin and and Hassel–Varley). The blow up phenomenon at finite time in spatial cases have been discussed for all the model systems. For each model system, mathematical conditions are established under which species can blow up at finite time. Spatio-temporal dynamics and weakly nonlinear analysis have been thoroughly studied. Stability analysis of these model systems have been investigated and concentrated on discussing the Turing patterns. Theoretical analysis of the patterns formations has been described by using amplitude equations and the results are validated by numerical simulations. Mathematical computations of the structural models have been studied and explored their blow up phenomena under the effects of diffusion with the pattern formation of preys and predator populations in the spatiotemporal domain.The best chance of survival of the species upon diffusion has been discussed elaborately.