In this work, an attempt is made to understand the dynamics of a modified Leslie-Gower model with nonlinear harvesting and Holling type-IV functional response. We study the model system using qualitative analysis, bifurcation theory and singular optimal control. We show that the interior equilibrium point is locally asymptotically stable and the system under goes a Hopf bifurcation with respect to the ratio of intrinsic growth of the predator and prey population as bifurcation parameter. The existence of bionomic equilibria is analyzed and the singular optimal control strategy is characterized using Pontryagin's maximum principle. The existence of limit cycles appearing through local Hopf bifurcation and its stability is also examined and validated numerically by computing the first Lyapunov number. Optimal singular equilibrium points are obtained numerically for various discount rates.

In this paper, we study the complex dynamics of a spatial nonlinear predator-prey system under harvesting. A modified Leslie-Gower model with Holling type IV functional response and nonlinear harvesting of prey is considered. We perform a detailed stability and Hopf bifurcation analysis of the spatial model system and determine the direction of Hopf bifurcation and stability of the bifurcating periodic solutions. Numerical simulations were performed to figure out how Turing patterns evolve under nonlinear harvesting. Simulation study leads to a few interesting sequences of pattern formation, which may be relevant in real world situations.

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In this paper, a spatial model has been designed to study a damaged diffusive eco-epidemiological system of Tilapia and Pelican populations in Salton Sea, California, USA. The nature of different equilibrium points and the existence of Hopf bifurcations are obtained. Conditions for Turing instability caused by local random movement of populations are derived. Numerically, the presence/existence of the wave of chaos phenomena is reported. Further, we show that the contact rate between susceptible and infected Tilapia population plays an important role in the distribution of the infected Tilapia population. The results suggest that the removal of infected Tilapia at regular time duration and controlling salinity will help to restore the system which provides a perspective for conservation strategy.

In this paper, we consider a Gause-type model system consisting of two prey and one predator. Gestation period is considered as the time delay for the conversion of both the prey and predator. Bobcats and their primary prey rabbits and squirrels, found in North America and southern Canada, are taken as an example of an ecological system. It has been observed that there are stability switches and the system becomes unstable due to the effect of time delay. Positive invariance, boundedness, and local stability analysis are studied for the model system. Conditions under which both delayed and nondelayed model systems remain globally stable are found. Criteria which guarantee the persistence of the delayed model system are derived. Conditions for the existence of Hopf bifurcation at the nonzero equilibrium point of the delayed model system are also obtained. Formulae for the direction, stability, and period of the bifurcating solution are conducted using the normal form theory and center manifold theorem. Numerical simulations have been shown to analyze the effect of each of the parameters considered in the formation of the model system on the dynamic behavior of the system. The findings are interesting from the application point of view.