A General Model of Population Dynamics Accounting for Multiple Kinds of Interaction

Abstract

Population dynamics has been modelled using differential equations almost since Malthus times, more than two centuries ago. Basic ingredients of population dynamics models are typically a growth rate, a saturation term in the form of Verhulst’s logistic brake, and a functional response accounting for interspecific interactions. However, intraspecific interactions are not usually included in the equations. The simplest models use linear terms to represent a simple picture of the nature; meanwhile, to represent more complex landscapes, it is necessary to include more terms with a higher order or that are analytically more complex. The problem to use a simpler or more complex model depends on many factors: mathematical, ecological, or computational. To address it, here we discuss a new model based on a previous logistic-mutualistic model. We have generalized the interspecific terms (for antagonistic and competitive relationships), and we have also included new polynomial terms to explain any intraspecific interaction. We show that, by adding simple intraspecific terms, new free-equilibrium solutions appear driving a much richer dynamics. These new solutions could represent more realistic ecological landscapes by including a new higher order term.

Prevalence of mutualism in a simple model of microbial co-evolution

“…In this section we will introduce what might be the simplest possible model capable of exhibiting evolutionary transitions between different ecological regimes. We will consider two microbial species whose populations evolve in time according to the generalised logistic model ( Stucchi et al, 2020 ), briefly described in Box 1 . Specialising this model for just two species, and ignoring intraspecific cooperation or direct competition (i.e., b ii = 0), the two equations that describe this minimal ecological community are This system of differential equations is able to describe every kind of ecological interaction.…”

“…The idea behind the population model of Stucchi et al (2020 ) is to extend Velhurst’s logistic equations of populations by making the parameters and ā i to depend on the interactions with the environment as well as the population sizes of all species in the community as Here r i is the vegetative growth rate of species i, b ik is the rate of benefit (if positive) or hindrance (if negative) on species i due to the interaction with species k , and p is the total number of species in the ecosystem. The coefficients a i measure intraspecific competitions (hence a i > 0) due to a limitation of the environmental resources.…”

“…According to Stucchi et al (2020 ), the linear stability of the finite stationary solution for two species can be analyzed from the Jacobian matrix …”

“…The model relies upon the availability of a phenomenological model of ecological interactions ( García-Algarra et al, 2014; Stucchi et al, 2020 ) that is capable of describing mutualistic, competitive, and antagonistic interactions with a simple tuning of the parameters—in a way that generalized Lotka-Volterra models are not capable of. The clue in devising our present evolutionary model has been linking those parameters to microscopic interactions between the species involved— which we have chosen as microbes for the sake of simplicity.…”

“…The addition of Holling-type functional responses ( Wright, 1989 ) is the usual way to control an unbounded growth of the populations, at the expense of rendering the model analytically intractable. Recently though, a new population model has been introduced in which both, species’ vegetative growth and interspecific interactions, are limited via logistic terms ( García-Algarra et al, 2014; Stucchi et al, 2020 ). The resulting equations are amenable to analytic treatment and, at the same time, provide sensible results whether the interactions are of antagonistic or mutualistic nature.…”