  The strong Allee effect plays an important role on the evolution of
population in ecological systems. One important concept is the Allee threshold
that 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 a
species since when the initial data is below the Allee threshold the population
tends to extinction, rather than persistence. Another interesting feature of
population evolution is that a species whose movement strategy follows a
conditional dispersal strategy is more likely to persist. In other words, the
biased movement can be a benefit for the persistence of the population. The
coexistence of the above two conflicting mechanisms makes the dynamics rather
intricate. However, some numerical results obtained by Cosner et. al. (SIAM J.
Appl. Math., Vol. 81, No. 2, 2021) show that the directed movement can
invalidate the strong Allee effect and help the population survive. To study
this intriguing phenomenon, we consider the pattern formation and local
dynamics for a class of single species population models of that is subject to
the strong Allee effect. We first rigorously show the existence of multiple
localized solutions when the directed movement is strong enough. Next, the
spectrum analysis of the associated linear eigenvalue problem is established
and used to investigate the stability properties of these interior spikes. This
analysis proves that there exists not only unstable but also linear stable
steady states. Then, we extend results of the single equation to coupled
systems, and also construct several non-constant steady states and analyze
their stability. Finally, numerical simulations are performed to illustrate the
theoretical results.

Analysis of PDEs