  Mathematical models of spatial population dynamics typically focus on the
interplay between dispersal events and birth/death processes. However, for many
animal communities, significant arrangement in space can occur on shorter
timescales, where births and deaths are negligible. This phenomenon is
particularly prevalent in populations of larger, vertebrate animals who often
reproduce only once per year or less. To understand spatial arrangements of
animal communities on such timescales, we use a class of diffusion-taxis
equations for modelling inter-population movement responses between $N \geq 2$
populations. These systems of equations incorporate the effect on animal
movement of both the current presence of other populations and the memory of
past presence encoded either in the environment or in the minds of animals. We
give general criteria for the spontaneous formation of both stationary and
oscillatory patterns, via linear pattern formation analysis. For $N=2$, we
classify completely the pattern formation properties using a combination of
linear analysis and non-linear energy functionals. In this case, the only
patterns that can occur asymptotically in time are stationary. However, for $N
\geq 3$, oscillatory patterns can occur asymptotically, giving rise to a
sequence of period-doubling bifurcations leading to patterns with no obvious
regularity, a hallmark of chaos. Our study highlights the importance of
understanding between-population animal movement for understanding spatial
species distributions, something that is typically ignored in species
distribution modelling, and so develops a new paradigm for spatial population
dynamics.
