Population dynamics under climate change: persistence criterion and effects of fluctuations.

The present paper is devoted to the investigation of population dynamics under climate change. The evolution of species is modelled by a reaction-diffusion equation in a spatio-temporally heterogeneous environment described by a climate envelope that shifts with a time-dependent speed function. For a general almost-periodic speed function, we establish the persistence criterion in terms of the sign of the approximate top Lyapunov exponent and, in the case of persistence, prove the existence of a unique forced wave solution that dominates the population profile of species in the long run. In the setting for studying the effects of fluctuations in the shifting speed or location of the climate envelope, we show by means of matched asymptotic expansions and numerical simulations that the approximate top Lyapunov exponent is a decreasing function with respect to the amplitude of fluctuations, yielding that fluctuations in the shifting speed or location have negative impacts on the persistence of species, and moreover, the larger the fluctuation is, the more adverse the effect is on the species. In addition, we assert that large fluctuations can always drive a species to extinction. Our numerical results also show that a persistent species under climate change is invulnerable to mild fluctuations, and becomes vulnerable when fluctuations are so large that the species is endangered. Finally, we show that fluctuations of amplitude less than or equal to the speed difference between the shifting speed and the critical speed are too weak to endanger a persistent species.

Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?

The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system, [Formula: see text]where [Formula: see text], a, b, [Formula: see text], and [Formula: see text] are positive constants. Assume [Formula: see text] . Then if in addition [Formula: see text] holds, it is proved that [Formula: see text] is the spreading speed of the solutions of (0.1) with nonnegative continuous initial function [Formula: see text] with nonempty compact support, that is, [Formula: see text]and [Formula: see text]where [Formula: see text] is the unique global classical solution of (0.1) with [Formula: see text]. It is also proved that, if [Formula: see text] and [Formula: see text] holds, then [Formula: see text] is the minimal speed of the traveling wave solutions of (0.1) connecting (0, 0) and [Formula: see text], that is, for any [Formula: see text], (0.1) has a traveling wave solution connecting (0, 0) and [Formula: see text] with speed c, and (0.1) has no such traveling wave solutions with speed less than [Formula: see text]. Note that [Formula: see text] is the spatial spreading speed as well as the minimal wave speed of the following Fisher-KPP equation, [Formula: see text]Hence, if [Formula: see text] and [Formula: see text], or [Formula: see text] and [Formula: see text], then the chemotaxis neither speeds up nor slows down the spatial spreading in (0.1).