Deterministic Mathematical Models in Population Ecology.

We formulate a three species prey–predator model supplying allochthonous inputs to the top predator in the presence of top predator harvesting. All the equilibria of the system are determined and their stability nature are investigated. With the help of Pontryagin’s maximum principle, fishing effort is used as an optimal controller to investigate the optimal solution. The system dynamics for different values of preference parameter (
 $$c$$
 ), catchability coefficient (
 $$q$$
 ), allochthonous inputs (
 $$A$$
 ) and harvesting effort (
 $$E$$
 ) are presented. Bifurcation analysis is done with respect to harvesting and existence of chaos, 2-cycle, 4-cycle, limit cycle and steady state behaviours are observed. The existence of Hopf bifurcation, limit point, limit point cycle, period doubling and branch point are observed in the system. The dynamics of the system in the A–E parameter plane is presented. We observe that predator extinction risk decreases with the increasing of available allochthonous resources in the presence of harvesting. Our results suggest that allochthonous inputs at high levels can stabilize food webs and remove the possibility of species extinction in the presence of harvesting.

Effects of Allochthonous Resources in a Three Species Food Chain Model with Harvesting

The main purpose of this article is to consider a Lotka–Volterra predator–prey system with ratio-dependent functional responses and feedback controls. By using a comparison theorem and constructing a suitable Lyapunov function as well as developing some new analysis techniques, we establish a set of easily verifiable sufficient conditions which guarantee the permanence of the system and the global attractivity of a positive solution for the predator–prey system. Furthermore, some conditions for the existence, uniqueness, and stability of a positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis method. In additional, some numerical solutions of the equations describing the system are given to verify that the obtained criteria are new, general, and easily verifiable.

/pdf/well-posedness-of-a-ratio-dependent-lotka-volterra-system-23xsi9d2hg.pdf

Well-posedness of a ratio-dependent Lotka–Volterra system with feedback control

By using a fixed point theorem in cones, some sufficient conditions are established for the existence of three positive periodic solutions for a kind of nonlinear neutral functional differential equations with impulse and state dependent delays.

Existence of three positive periodic solutions of nonlinear neutral functional differential equations with impulses and delays

In this paper, by using Mawhin's continuation theorem of coincidence degree theory, we establish the existence of at least four positive periodic solutions for a discrete time Lotka-Volterra competitive system with harvesting terms. An example is given to illustrate the effectiveness of our results.

Four positive periodic solutions of a discrete time Lotka-Volterra competitive system with harvesting terms

In this paper, by using Mawhin’s continuation theorem of coincidence degree theory, we establish the existence of at least two positive periodic solutions for a discrete time delayed predator–prey system with harvesting terms An example is given to illustrate the effectiveness of our results

Multiple Positive Periodic Solutions of a Discrete Time Delayed Predator–Prey System with Harvesting Terms

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Four positive periodic solutions of a discrete time Lotka-Volterra competitive system with harvesting terms

Multiple Positive Periodic Solutions of a Discrete Time Delayed Predator–Prey System with Harvesting Terms