The logistic equation is the simplest way to limit the growth in population dynamics. When ecological interactions come into play different types of models can be found, e.g. for consumer-resource systems the generalized Lotka–Volterra equation is widely used to model direct competition or predator–prey systems, and for mutualistic systems a functional response that limits the unbounded growth due to the mutual benefit is needed (usually Holling's type II). Based on a recent general model of population dynamics with intraspecific interactions we present a factored general logistic model of population dynamics with inter- and intraspecific interactions. A major advantage of this model is that it can be used for any type of interspecific ecological interaction and also for beneficial or detrimental intraspecific interaction, and always in a bounded way. In this study we write a general logistic model in a factored way to obtain the stationary solutions by a system of simple linear equations and we formulate the analytical expression for the Jacobian matrix of all the stationary solutions for an arbitrary number of populations. We also show that this simple model can be used to represent complex ecological systems; as an illustration we study some examples such as a stable direct competition with intraspecific cooperation, a predator–prey system with cooperative preys, a mutualism with harmful intraspecific interactions and a real bacterial system with 4 populations.
Bibliographical noteFunding Information:
We thanks the anonymous reviewer and Christopher Moore for their useful comments that enriched the manuscript. This work was supported by Ministry of Education, Culture, and Sport of Spain ( PGC2018-093854-B-100 ).
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- Ecological interaction
- Intraspecific interaction
- Logistic equation
- Population dynamics
- Verhulst model