The book is concerned with various models arising from the study of the dynamics of the population facing climate change ranging from fixed to free boundary domain. The original model has been proposed by Berestycki et al. [13] and then it has been attracted by many mathematicians to develop in various directions and contexts. The content of this book comprises six chapters as follows:
Chapter 1: We present some preliminary results that are used in this book.
Chapter 2 deals with the problem in an infinite cylindrical domain and in the whole space where the reaction term is (resp.) independent or periodically dependent on time. The novelty of this work is that we consider a global condition in terms of the spectral theory to express that the environment of the population is globally unfavorable at infinity instead of compactly favorable as in [13, 27, 28]. We further study the concentration of the species in the cylindrical domain when the exterior domain is changed to be extremely unfavorable.
Chapter 3, we provide a spectral condition to prove a Liouville-type result for an elliptic equation in a periodic shear flow with a very general nonlinearity of Fisher-KPP type. The main tool of this result is the maximum principle in the whole space RN obtained by Berestycki and Rossi in a recent well-quoted work [29].
Chapter 4 focuses on finding the sharp criterion for the existence, nonexistence and uniqueness of a positive solution of a fully semilinear elliptic equation. When the divergence of the drift term is zero, the existence of a positive solution can be characterized by the amplitude of the drift term under some fair assumptions on the growth rate. The large time behavior of the associated parabolic equation is considered, where we have to deal with the case of possibly unbounded coefficients.
Chapter 5, we extend the existence, nonexistence and uniqueness in the second chapter for a quasilinear equation involving a p-Laplacian operator. The main difficulty is that it seems hard to apply the strong maximum principle and thus we make use of a variational approach to attain an important comparison principle in the whole space, which is the key idea in our technique to derive our expected results.
Chapter 6, we study a free boundary problem with two competing species in the Lotka-Volterra system under the impact of climate change and seasonal effect in the left-shifting environment. We prove two main results: (i) The vanishing phenomena of both inferior and superior competitors when an unfavorable environment caused by an external effect spreads at a speed faster than the spreading speed of the inferior competitor; (ii) The stability of the trichotomy vanishing-spreading-transition of the inferior competitor while the superior one always loses in the competition.
Chapter 7, we further study the impact of climate change on the global dynamics of mosquitoes, we consider a reaction-diffusion free boundary model with conditional dispersal in a heterogeneous environment. The main concern in this chapter is to study long-time dynamics of solutions assuming that the environment is globally unfavorable determined by a spectral condition at infinity and the species is assumed to be losing in the spreading a long their boundary of its habitat.
