Qualitative And Quantitative Analyses Of Prey-predator Population Models With Individual Size Differences

Biomathematics is an interdisciplinary subject between biology and mathematics, which solvesbiological problems by means of mathematical approaches, and researches mathematical problemswith biological backgrounds. Population is an essential unit of research in biology, which is made ofindividuals living in a certain habitat, and has the potential of reproduction, independent structures,common characteristics and biological functions. Population is one of the basic existence patterns ofspecies of nature. In order to maintain ecological balance, the biodiversity and make full use of therenewable biological resources, we must study in-depth the laws of the evolution ofbiological populations. Thus, scholars over the world have established a large number ofmathematical models. The changing rule of solutions to the models predicts the evolutional trends ofthe populations. The mathematical modelling with size-structure is an important approach. Here, sizemeans the body size of individuals, refers to a continuous index corresponding to an individual, suchas weight, length, diameter, volume and maturity. Predating refers to the phenomenon that a specieseats another, which is common in nature. Prey-predator ship plays a significant role in the evolution ofevolved populations.Firstly, a prey-predator model based on size distributions is formulated, which is aninitial-boundary value problem of partial and integro-differential equations. Then the followingproblems are carefully investigated: the existence, uniqueness, nonnegativity of solutions, theformal solutions of the model, existence and local stability of the equilibria, and some numericalsimulations are presented. Some theoretical results are obtained by the use of differentialequations, functional analysis and other tools, which supply a solid ground for the practicalapplication of the model.The research works in the dissertation are made up of two parts: the first part is in chapter 2, andthe second part in chapter 3. The outline is as follows:The chapter 2 studies the well-posedness of the size-structured prey-preadator populations model,including the existence, boundedness, nonnegativity and uniqueness of the solutions. Firstly, as thetwo size variables result in the inconvenience, we make a biological assumption that the predator sizeis greater than or equal to that of the prey, and process the relevant parameters to get a unified model.Then we prove the boundedness of solutions of the system by the application of the linear comparisonprinciple. Secondly, we prove the existence, uniqueness and non-negativity of solutions of the systemby difference approximation method.The chapter 3 is devoted to the formal solutions of the size-structured prey-predator model, and existence and local stability of the equilibria of the model. Firstly the formal solutions are derived bycharacteristic curves. Then we analyze the existence, local stability of three kinds of equilibriumsolutions of the model. Finally, the corresponding numerical simulations about the local stability ofequilibria of the model are made by using C# and Excel tools, which display intuitively the stabilityresults.