Does Random Dispersion Help Survival?

Abstract

Many species live in colonies that prosper for a while and then collapse. After the collapse the colony survivors disperse randomly and found new colonies that may or may not make it depending on the new environment they find. We use birth and death chains in random environments to model such a population and to argue that random dispersion is a superior strategy for survival.Keywords Stochastic model · Random environment · Survival strategy The ModelMany species (such as ants or early humans) survive according to the following pattern. A few individuals start a colony, the colony prospers until it collapses due to exhaustion of resources or some external factor, a few individuals survive the collapse, disperse randomly and found new colonies. The pattern starts over. In this paper we are interested in modeling this process and comparing it to other survival strategies. Similar models have been used for related questions in population biology, see [1,2].Next we give an informal description of our stochastic model. We will be more precise in the proof of Theorem 1.The individuals are divided in independent colonies. Each colony is a birth and death chain with birth rate λ sampled from a fixed distribution μ and death rate 1. Moreover, each colony is associated with a random time T sampled from a fixed distribution ν. At time T each individual of the colony starts its own independent colony. Each new colony has a new λ and a new T sampled from μ and ν, respectively. The new (λ, T ) is independent of everything else but λ and T need not be independent.We are particularly interested in distributions μ for which μ([0, 1)) is almost 1. That is, most of the λ's that are sampled are below 1 and the new colony collapses very quickly

Colonization and Collapse

“…Colonies collapse independently of each other at rate a > 0. The proof in Schinazi (2015) may be adapted to show that survival has positive probability if and only if…”

Dispersion as a Survival Strategy

“…Individuals of these populations disperse, trying to make new colonies that may succeed settling down depending on the new environment they encounter. Recently, Schinazi [9] and Machado et al [7] proposed stochastic models for this kind of population dynamics. For these models they concluded that dispersion is a good survival strategy.…”

“…Based on the model proposed by Artalejo et al [1], and adapting some ideas from Schinazi [9] and Machado et al [7], we analyze growth models of populations subject to disasters, where after the collapse species adopt dispersion as a survival strategy. We show that dispersion is not always a good strategy to avoid the population extinction.…”

Modelagem estocástica para dinâmicas de colonização e colapso

“…Esses indivíduos se dispersam tentando fazer novas colônias que podem ou não se estabelecer dependendo do ambiente que encontram. Recentemente, Schinazi [20] propôs um modelo estocástico não espacial para modelar tal tipo de dinâmica populacional e mostrou que a dispersão aleatóriaé uma boa estratégia para a sobrevivência da população. O modelo e os principais resultados em Schinazi [20] estabelecem o seguinte.…”

“…Recentemente, Schinazi [20] propôs um modelo estocástico não espacial para modelar tal tipo de dinâmica populacional e mostrou que a dispersão aleatóriaé uma boa estratégia para a sobrevivência da população. O modelo e os principais resultados em Schinazi [20] estabelecem o seguinte. Teorema 1.2.1 (Teorema 1 em [20]).…”