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The strong Allee effect plays an important role on the evolution ofpopulation in ecological systems. One important concept is the Allee thresholdthat determines the persistence or extinction of the population in a long time.In general, a small initial population size is harmful to the survival of aspecies since when the initial data is below the Allee threshold the populationtends to extinction, rather than persistence. Another interesting feature ofpopulation evolution is that a species whose movement strategy follows aconditional dispersal strategy is more likely to persist. In other words, thebiased movement can be a benefit for the persistence of the population. Thecoexistence of the above two conflicting mechanisms makes the dynamics ratherintricate. However, some numerical results obtained by Cosner et. al. (SIAM J.Appl. Math., Vol. 81, No. 2, 2021) show that the directed movement caninvalidate the strong Allee effect and help the population survive. To studythis intriguing phenomenon, we consider the pattern formation and localdynamics for a class of single species population models of that is subject tothe strong Allee effect. We first rigorously show the existence of multiplelocalized solutions when the directed movement is strong enough. Next, thespectrum analysis of the associated linear eigenvalue problem is establishedand used to investigate the stability properties of these interior spikes. Thisanalysis proves that there exists not only unstable but also linear stablesteady states. Then, we extend results of the single equation to coupledsystems, and also construct several non-constant steady states and analyzetheir stability. Finally, numerical simulations are performed to illustrate thetheoretical results.

Existence and Stability of Localized Patterns in the Population Models with Large Advection and Strong Allee Effect