Entire Solutions Of Integro-difference Equations

In this thesis,we discuss the entire solutions for the following integro-difference e-quation un+1(x)= ?R k(x-y)f(un(y))dy,x? R,n = 0,1,2...,where un(x)denotes the population density at time n and at the position x in population dynamics,f:R+?R+ is the so-called birth function,k formulates the migration probability of the individuals.Firstly,we consider the entire solutions of the integro-difference equation under the monostable condition when the birth function is monotone.Using the linearized charac-teristic equation at the 0 and Ikehara's theorem,we obtain the asymptotic behavior of the monotone traveling wave solutions,which implies that the traveling wave solutions with non-critical wave speeds exponentially decay while the traveling wave solutions with critical wave speed do not exponentially decay at negative infinity.Applying the decay estimation and comparison principle,we confirm the existence as well as the continuous dependence on the parameters of entire solutions.Secondly,we investigate the existence of the entire solutions when the birth function is non-monotonic under monostable conditions.By constructing two monotonic functions admitting the same formulations near 0,we can construct generalized upper and lower solutions of the non-monotonic equation.By the comparison principle and the method of lower and upper solutions,we obtain the existence of entire solutions.Lastly,we study the entire solutions if the equation is bistable.By using the linearized characteristic equations at the 0 and 1,we obtain the prior decay rate of monotone traveling wave solution at—? and +?,which shows that the traveling wave solutions decay exponentially at—? and +?.By considering the interactions between two monotone traveling wave solutions with opposite propagation directions,we obtain the existence and some qualitative properties of entire solutions.