Beyond the pair approximation: Modeling colonization population dynamics

Vacancies in growing habitats promote the evolution of cooperation

We study evolutionary game dynamics in a growing habitat with vacancies. Fitness is determined by the global effect of the environment and a local prisoner's dilemma among neighbors. We study population growth on a one-dimensional lattice and analyze how the environment affects evolutionary competition. As the environment becomes harsh, an absorbing phase transition from growing populations to extinction occurs. The transition point depends on which strategies are present in the population. In particular, we find a 'cooperative window' in parameter space, where only cooperators can survive. A mutant defector in a cooperative community might briefly proliferate, but over time naturally occurring vacancies separate cooperators from defectors, thereby driving defectors to extinction. Our model reveals that vacancies provide a strong boost for cooperation by spatial selection.

Laws of Spatially Structured Population Dynamics on a Lattice

We consider spatial population dynamics on a lattice, following a type of a contact (birth–death) stochastic process. We show that simple mathematical approximations for the density of cells can be obtained in a variety of scenarios. In the case of a homogeneous cell population, we derive the cellular density for a two-dimensional (2D) spatial lattice with an arbitrary number of neighbors, including the von Neumann, Moore, and hexagonal lattice.are not allowed in the abstract, please move them to the main text, please remember to check the order of references in the full text after revision. We then turn our attention to evolutionary dynamics, where mutant cells of different properties can be generated. For disadvantageous mutants, we derive an approximation for the equilibrium density representing the selection–mutation balance. For neutral and advantageous mutants, we show that simple scaling (power) laws for the numbers of mutants in expanding populations hold in 2D and 3D, under both flat (planar) and range population expansion. These models have relevance for studies in ecology and evolutionary biology, as well as biomedical applications including the dynamics of drug-resistant mutants in cancer and bacterial biofilms.

Impact of colonization history on the composition of ecological systems

Observational studies of ecological systems have shown that different species compositions can arise from distinct species arrival orders during community assembly-also known as colonization history. The presence of multiple interior equilibria in the positive orthant of the state space of the population dynamics will naturally lead to history dependency of the final state. However, it is still unclear whether and under which conditions colonization history will dominate community composition in the absence of multiple interior equilibria. Here, by considering that only one species can invade at a time and there are no recurrent invasions, we show clear evidence that the colonization history can have a big impact on the composition of ecological systems even in the absence of multiple interior equilibria. In particular, we first derive two simple rules to determine whether the composition of a community will depend on its colonization history in the absence of multiple interior equilibria and recurrent invasions. Then we apply them to communities governed by generalized Lotka-Volterra (gLV) dynamics and propose a numerical scheme to measure the probability of colonization history dependence. Finally, we show, via numerical simulations, that for gLV dynamics with a single interior equilibrium, the probability that community composition is dominated by colonization history increases monotonically with community size, network connectivity, and the variation of intrinsic growth rates across species. These results reveal that in the absence of multiple interior equilibria and recurrent invasions, community composition is a probabilistic process mediated by ecological dynamics via the interspecific variation and the size of regional pools.