Traveling Waves For A Single Species With Age Structure On Two-Dimension Lattice

Lattice differential equations, namely infinite systems of ordinary differential equations indexed by points on a spatial lattice, for example the D-dimensional integer lattice ZD, have been proposed as models in various contents, such as ma-terial science, image processing, chemical kinetics, biology, etc. Numerical analysis becomes another important background of lattice differential equations with the advanced development of computers. Compared with the continuous partial dif-ferential equations, lattice differential equations have been found to exhibit much more complicated and colorful dynamics. Traveling wave solutions are one of the important topics in the study of lattice differential equations and has many research accomplishments so far. This paper is focused on the traveling wave solutions of a lattice model for a single-species with age-structure (immature and mature) in a two-dimensional patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay. The main content is divided into four chapters.Firstly, if the immature population is non-mobile, asymptotic stability of trav-eling wavefronts of this model is considered under the assumption that the nonlinear term satisfies the monostable condition. We prove that the traveling wavefront is exponentially stable by means of weighted energy method, when the initial pertur-bation around the wave is suitable small in a weighted norm. In fact, our results in this chapter can be extended smoothly to the general delayed Fisher-KPP dynamical system in 2-D lattice.Then, the existence of fast traveling waves connecting an equilibrium and a periodic orbit of this model is studied, under the assumption that the correspond-ing ODEs have heteroclinic orbits connecting an equilibrium point and a periodic solution. Our approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain Banach space coupled with an index formula of the associated Fredholm operator and some careful estimations of the nonlinear perturbation to ensure the existence of traveling waves connecting an equilibrium and a periodic orbit with large wave speeds in a neighborhood of a hete-roclinic connecting orbit of a corresponding ordinary delay differential equation. At the same time, we can still obtain the existence of periodic traveling waves around the uniform equilibrium by using the same method.In the third part, if the delay is small enough, the persistence of traveling waves of this model is considered under the bistable assumption. The known results demonstrate that the existence of traveling waves of lattice differential equations with nonlocal delay is completely similar to that without delay, when the quasi-monotone condition is satisfied. While the quasi-monotone condition is failure and the delay is large, some complexities appear for traveling waves, such as the periodic traveling waves around the uniform equilibrium as mentioned in chapter 3. In the fourth chapter, based on the exact estimation of Green function of the correspond-ing mixed type functional differential equations coupled with Fredholm alternative theorem proposed by Mallet-Paret, we prove the persistence of traveling waves with non-zero speed in the case of small delay under the bistable assumption, i.e. "small delay is harmless", in the sense that traveling waves of the equations under consid-eration persist when the small delay is incorporated in such an equation.There are many spacial heterogeneities in the natural environment, a typi-cal example of which is a certain periodicity. Finally, traveling waves in periodic monostable case is investigated. The existence and monotonicity of traveling waves is proved through constructing suitable sub-sup solutions, after that, the asymp-totic stability and uniqueness of traveling waves is derived by virtue of squeezing technique based on the comparison principle.