Asymptotic spreading for a time-periodic predator-prey system

Abstract

This paper is concerned with asymptotic spreading for a timeperiodic predator-prey system where both species synchronously invade a new habitat. Under two different conditions, we show the bounds of spreading speeds of the predator and the prey, which is proved by the theory of asymptotic spreading of scalar equations, comparison principle and generalized eigenvalue. We show either the predator or the prey has a spreading speed that is determined by the linearized equation at the trivial steady state while the spreading speed of the other also depends on the interspecific nonlinearity. From the viewpoint of population dynamics, our results imply that the predator may play a negative effect on the spreading of the prey while the prey may play a positive role on the spreading of the predator. t→∞ sup |x|>(c * + )t u(x, t) = 0 for any given > 0,u(x, t) > 0 for any given ∈ (0, c * ).Clearly, the definition of spreading speed states that if an observer were to move to the right or left at a fixed speed greater (less) than c * , then the local population density u(x, t) would eventually look like 0 (greater than 0) [41], and it also describes the speed at which the geographic range of the new population expands in population dynamics. In the case of monotone semiflows and cooperative systems, 2000 Mathematics Subject Classification. Primary: 35K45, 35K57; Secondary: 92D25.

Spatiotemporal propagation of a time‐periodic reaction–diffusion SI epidemic model with treatment

“…use spatiotemporal translation skills to deal with the difficulties caused by the nonautonomy. For other relevant works involving periodic equations or systems without quasi‐monotonicity, one can also refer to Wang and Lin 22 . and Lin 23 …”

“…For other relevant works involving periodic equations or systems without quasi-monotonicity, one can also refer to Wang and Lin. 22 and Lin. 23 On the other hand, the traveling wave solution is another powerful theoretical tool to model the geographic invasion and spread of a newly introduced epidemic.…”

Invasion waves for a nonlocal dispersal predator-prey model with two predators and one prey

“…In population dynamics, predator-prey systems have been widely studied due to their importance. Traveling wave, as a special solution maintaining its shape and moving at a constant speed, is a very important dynamical issue in the field of reaction-diffusion equations and is a useful tool to predict some ecological phenomena of different predator-prey systems, one can see Ai et al [2], Du et al [9], Hsu and Lin [14], Lin [19], Wang and Lin [25], Zhang and Jin [34], Zhang et al [35] for further understanding. Moreover, Lin and Ruan [20] used the idea of contracting rectangles to verify the limit behavior of a traveling wave solution.…”