Nonlinear Modelling of Competitive Ecosystem

This paper deals with the mathematical modelling of an ecosystem where two or more species are competing for natural resources. Using fixed point and phase portrait analysis, the linear and non-linear models of the system are studied. Fixed-point analysis is used to find the stability conditions and phase portrait analysis is used to study the direction of flow in which system evolves. The phase portraits are plotted using MATLAB programmes. For a linear model, there exists a coexistence and catastrophe phase. In the coexistence phase for linear model, it is observed that the one species is dominant on the other species and vice versa in terms of population by shifting of dominance. The limit cycle in the phase portrait concludes that conservation of total population of the ecosystem exists. Also, in some cases, the catastrophe is observed which concludes that the total population of ecosystem gets extinct despite of transfer of dominance amongst the species. For nonlinear model, the intra- and inter-species interactions between the species based on Lotka Volterra model is considered. If an intermediate is introduced in the system then the whole dynamics of the system changes with the arising of chaotic state in which sustenance and extinction of the species is uncertain. It is observed that stability transfers from one equilibrium point to another with the change of parameter values. It is concluded that if the parameters are maintained effectively then for any kind of interactions between the photosynthetic algal species, the species will exhibits coexistence behaviour within a limited range and effectively utilise the natural resources from the environment which may be useful in conserving the biological diversity and the ecological cycles in environment.