Studying Both Direct and Indirect Effects in Predator-Prey Interaction

Studying and modelling the interaction between predators and prey have been one of the central topics in ecology and evolutionary biology. In this thesis, we study two different aspects of predator-prey interaction: direct effect and indirect effect. Firstly, we study the direct predation between predators and prey in a patchy landscape. Secondly, we study indirect effects between predators and prey. Thirdly, we extend our previous model by incorporating a stage-structure into prey. Finally, we further extend our previous model by incorporating spatial structures into modelling

Global attractivity of a positive periodic solution for a nonautonomous stage structured population dynamics with time delay and diffusion

AbstractBy employing the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for a nonautonomous stage structured population dynamics with time delay and diffusion is established. Further, by constructing a Lyapunov functional and using the result of the existence of positive periodic solution, the attractivity of a positive periodic solution for above system is obtained

Predator-prey-subsidy population dynamics on stepping-stone domains with dispersal delays

We examine the role of the travel time of a predator along a spatial network on predator-prey population interactions, where the predator is able to partially or fully sustain itself on a resource subsidy. The impact of access to food resources on the stability and behaviour of the predator-prey-subsidy system is investigated, with a primary focus on how incorporating travel time changes the dynamics. The population interactions are modelled by a system of delay differential equations, where travel time is incorporated as discrete delay in the network diffusion term in order to model time taken to migrate between spatial regions. The model is motivated by the Arctic ecosystem, where the Arctic fox consumes both hunted lemming and scavenged seal carcass. The fox travels out on sea ice, in addition to quadrennially migrating over substantial distances. We model the spatial predator-prey-subsidy dynamics through a “stepping-stone” approach. We find that a temporal delay alone does not push species into extinction, but rather may stabilize or destabilize coexistence equilibria. We are able to show that delay can stabilize quasi-periodic or chaotic dynamics, and conclude that the incorporation of dispersal delay has a regularizing effect on dynamics, suggesting that dispersal delay can be proposed as a solution to the paradox of enrichment

A nonautonomous predator–prey system with stage structure and double time delays

AbstractIn the present paper we study a nonautonomous predator–prey model with stage structure and double time delays due to maturation time for both prey and predator. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the immature prey. Based on some comparison arguments we discuss the permanence of the species. By virtue of the continuation theorem of coincidence degree theory, we prove the existence of positive periodic solution. By means of constructing an appropriate Lyapunov functional, we obtain sufficient conditions for the uniqueness and the global stability of positive periodic solution. Two examples are given to illustrate the feasibility of our main results

Two-patch herbivore/vegetation models with density-dependent migration

In this thesis we constructed two mathematical models for herbivore/vegetation interactions in environment of two patches, using the metaphysiological approach and a density-dependent migrations. In the first model we considered the case when the environment is constant, and we constructed a system of four perturbed ordinary differential equations describing the dynamics when only herbivores allowed to move between the two patches searching for food. The model contain two different timescales, fast for migrations and slow for the other demographic changes in the system. We used the geometric singular perturbation theory in order to reduce the dimension of the system. Using the continuation software AUTO we provided bifurcation diagrams for the reduced systems and we also provided some numerical illustrations to show the dynamics of the system for different migrations propensities. We analyzed the bifurcation diagrams using Morse decompositions and Conley index theory, to confirm their correctness. We constructed a second mathematical model, by considering that the vegetation growth depends on seasonal rainfall and the soil moisture. We provided some numerical simulations to illustrate several variates of dynamics for different migration speed and, when the migration propensities and the vegetation quality are change

Permanence and extinction for a delayed periodic predator-prey system

In this paper, the permanence, extinction and periodic solution of a delayed periodic predator-prey system with Holling type IV functional response and stage structure for prey is studied. By means of comparison theorem, some sufficient and necessary conditions are derived for the permanence of the system

Mathematical Modelling of Ecological Systems in Patchy Environments

In this thesis, we incorporate spatial structure into different ecological/epidemiological systems by applying the patch model. Firstly, we consider two specific costs of dispersal: (i) the period of time spent for migration; (ii) deaths during the dispersal process. Together with the delayed logistic growth, we propose a two-patch model in terms of delay differential equation with two constant time delays. The costs of dispersal, by themselves, only affect the population sizes at equilibrium and may even drive the populations to extinction. With oscillations induced by the delay in logistic growth, numerical examples are provided to illustrate the impact of loss by dispersal. Secondly, we study a predator-prey system in a two-patch environment with indirect effect (fear) considered. When perceiving a risk from predators, a prey may respond by reducing its reproduction and decreasing or increasing (depending on the species) its mobility. The benefit of an anti-predation response is also included. We investigate the effect of anti-predation response on population dynamics by analyzing the model with a fixed response level and study the anti-predation strategies from an evolutionary perspective by applying adaptive dynamics. Thirdly, we explore the short-term or transient dynamics of some SIR infectious disease models over a patchy environment. Employing the measurements of reactivity of equilibrium and amplification rates previously used in ecology to study the response of an ecological system to perturbations to an equilibrium, we analyze the impact of the dispersals/travels between patches and other disease-related parameters on short term dynamics of these spatially structured disease models. This contrasts with most existing works on modelling the dynamics of infectious disease which are only interested in long-term disease dynamics in terms of the basic reproduction number

Periodic solutions of a discrete-time diffusive system governed by backward difference equations

A discrete-time delayed diffusion model governed by backward difference equations is investigated. By using the coincidence degree and the related continuation theorem as well as some priori estimates, easily verifiable sufficient criteria are established for the existence of positive periodic solutions