Spatial Dynamics Of Periodic Reaction-diffusion Model For Single Species With Age Structure

In the last few decades, the theory for travelling wave solutions and spreading speedfor nonlinear evolution system, especially for parabolic type system, has attracted lotsof attention. In population biology, this two concept have important significance andstrongly background. This thesis will focus on the existence of travelling waves andcomputing the spreading speed for given equation.Firstly, we establish a mathematical model on matured population under proper con-dition for a class of species which have periodic behaviour, it turn out to be a reaction-difusion equations with nonlocal delayed nonlinear reaction terms. And it used as pro-totypes to motivate appropriate abstract formulations of dynamic systems with unimodalnonlinearity.Secondly, we apply the theory of semigroups of linear operator to study the solutionof the equation with Cauchy initial value problem. And find the sufcient conditions forthe existence and uniqueness of classical solution, then we find the birth function can notguarantee a classical solution, but a mild solution and has countable non-diferentiablepoints. Also, for the kinetic dynamics, by reducing the Poincare′map to a map definedinR, we give sufcient condition for the unimodal birth function such that the equationgenerate monostable dynamical system and bistable dynamical system.Finally, we consider the existence of spreading speed and traveling waves for monos-table systems and bistable systems. For monostable system preserving monotonicity, bythe theory of monotone semiflow we prove existence of traveling wave and spreadingspeed, and give explicitly expression of spreading speed. And for non-monotone monos-table, we employ the idea that sandwiching the non-monotone equation in between t-wo constructed auxiliary monotone equations and developing a comparison principle toestablish the result of spreading speed,. For monotone bisable system, we investigatemonotone semiflow of bistable type to prove the existence of traveling waves.