Theory Of Impulsive Differential Equations With Application In Population Ecology

Impulsive differential equations are suitable for the mathematical modelling of evolutionary processes which experience a change of state abruptly owing to instantaneous perturbations. In terms of the mathematical treatment, the presence of impulses gives the system a mixed nature, both continuous and discrete.The theory of impulsive differential equations is much richer than the corresponding theory of differential equations without impulsive effects. In Chapter 1, the paper gives some fundamental theory of impulsive differential equations. In Chapter 2,3 and 4, by using the theories of continuous dynamics,impulsive dynamics, discrete dynamics and the methods of nonlinear analysis, mathematical simulation, the paper investigates the impulsive release strategies of epidemic models for pest control, the existences and stabilities for the periodic solutions of impulsive dynamical systems for one species , two species and N species with Gom-pertz law of groth. We investigate the optimal harvesting policies for periodic Gompertz ecosystem with impulsive harvest.In Chapter 2, first of all, we propose a single species model with impulsive diffusion between two patches, which provides a more natural description of population dynamics. By using the discrete dynamical system generated by a monotone, concave map for the population and a (?)₁ — (?)₂ variation we prove that the map always has a globally stable positive fixed point. This means that a single species system with impulsive diffusion always has a globally stable positive periodic solution. This result is further substantiated by numerical simulation. Besides, a seasonal seed dispersal mathematical model is presented and studed. Further numerical simulations show that the diffusion can save the extinction though the species has a negative growth rate in one patch. Lastly, the impulsive exploitation of single species modelled by periodic Gompertz ecosystem is studied. Choosing the maximum periodic biomass yield as the management objective, we investigate the optimal harvesting policies for periodic Gompertz ecosystem with impulsive harvest. When the optimal harvesting effort maximizes the periodic biomass yield, the corresponding optimal population level, and the maximum periodic biomass yield are obtained. In particular, it is proved that the maximum periodic biomass yield is in fact the maximum sustainable yield. The results extend and generalize the classical results of continuous harvest in renewable resources.In Chapter 3, a model concerning biologically-based impulsive control strategy for pest control is formulated and analyzed. It shows that there exists a globally stable...