Global Stability Of Interior Equilibrium Research Articles

Mathematical Modeling of Impact of Pollutants on Industries Based on Resource Biomass

Depletion of resources such as forestry, minerals etc. and resource-based industries such as wood and paper etc., due to rising pollution, is one of the biggest challenges which the humankind is facing today. In this paper, a mathematical model has been designed to give an insight into the effect of pollutants on natural resources which in turn affects the growth and stability of industries dependent on such biomass. The model is analyzed using stability theory of differential equations. Five dependent variables are considered in the model and some important assumptions are made. Two equilibria are found in the equilibrium analysis and conditions of local and global stability of interior equilibrium are obtained. Numerical simulation is also done to demonstrate the analytical findings. It is found in the study that as we impose an environmental tax on the polluters, the concentration of pollutants in the environment is controlled and the stable equilibrium shifts in such a way that the densities of resource biomass and dependent industries are close to the densities which correspond to the pollution free ecosystem.

Combine effects of square root functional response and prey refuge on predator–prey dynamics

ABSTRACT In this paper, a predator–prey interaction mathematical model has been developed incorporating prey refuge and square root functional response. It is considered that the prey species grow logistically in the absence of predator. Then, positivity and boundedness of solutions of our proposed system have been investigated. After that all possible equilibrium points of our proposed system are determined. Also, the local stability of our proposed system around the equilibrium points has been discussed. Global stability of interior equilibrium is also investigated. The Hopf bifurcation analysis has been done with respect to prey refuge rate (c), conservation rate of prey (b) and death rate of predator (d). Finally, some numerical-simulated results have been presented to test the validity of our theoretical findings.

Qualitative analysis of a diffusive Crowley–Martin predator–prey model: the role of nonlinear predator harvesting

In this paper, we discuss a diffusive predator–prey system with mutually interfering predator and nonlinear harvesting in predator with Crowley–Martin functional response. The mathematical analysis of the system starts with the existence and uniqueness of solution of the system using $$C_0$$ semigroup. The analysis reflects that the upper bound of rate of predator harvesting for the coexistence of the species can be guaranteed. In addition, we establish the existence and nonexistence of non-constant positive steady state. Explicit conditions on predator harvesting are obtained for local and global stability of interior equilibrium and also for the existence and nonexistence of non-constant steady-state solution. We also investigate conditions for Turing instabilities of the diffusive system analytically. Our results show that the effort of harvesting (g) provides a threshold value for existence of non-constant positive stationary solution. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the system exhibits interesting patterns. Some biological implications of obtained theoretical results have also been discussed.

Permanency in predator–prey models of Leslie type with ratio-dependent simplified Holling type-IV functional response

We consider a predator–prey model of Leslie type with ratio-dependent simplified Holling type-IV functional response. The novelty of the model is that the functional response simulates group defense of the prey kind, which in turn, affects permanency of the system and existence of limit cycles. We show that permanency of the system holds automatically for some values of parameters and provides sufficient conditions for global stability of interior equilibrium by constructing a Lyapunov function. We prove that for some values of parameters the system exhibits a Hopf cycle and provides conditions by which the corresponding stable Hopf cycle is the only cycle that model may have. Numerical simulations show that if the conditions are broken, the model may have more than one limit cycle, which is phenomenal among the predator–prey models with one interior equilibrium.