Ecological processes, such as reproduction, mobility, and interaction between species, play important roles in the maintenance of biodiversity. Classically, the cyclic dominance of species has been modelled using the nonhierarchical interactions among competing species, represented by the “Rock-Paper-Scissors” (RPS) game. Here we propose a cascaded channel model for analyzing the existence of biodiversity in the RPS game. The transition between successive generations is modelled as communication of information over a noisy communication channel. The rate of transfer of information over successive generations is studied using mutual information and it is found that “greedy” information transfer between successive generations may lead to conditions for extinction. This generalized framework can be used to study biodiversity in any number of interacting species, ecosystems with unequal rates for different species, and also competitive networks.

Ecological processes, such as reproduction, mobility, and interaction between species, have been shown to play important roles in the maintenance of biodiversity [

It has been shown that cyclic dominance alone is not sufficient to preserve biodiversity. When the interactions between individuals are local, spatially separated domains are dominated by one species form, leading to stable coexistence [

The primary motivation of this work is to model the information transfer between successive generations of evolution and link it to biodiversity. Further, we wish to explore the effect of the rate of information transfer on the existence of biodiversity. Yet another motivation is to study the nonhierarchical cyclic interactions between

The paper is organized as follows. Section

We consider the cyclic Lotka-Volterra model where three states

As the spatial version of the Lotka-Volterra model, we consider three subpopulations or species (

The stochastic model. The individuals of the three species are represented by

Let us model the transition from generation

The channel model. The lattice locations occupied by the individuals of the three species undergo transition from one generation to the next. Instead of labelling all the arrows, the transition probabilities have been listed towards the left of the species, for clarity. Note that

The transition probabilities are based on averaged probabilities (densities) ignoring the spatial structure. This is a mean-field approach, assuming a well-mixed population with the limit of population size

The channel transition probability matrix, for a well-mixed population, can be written as

From (

So far we have considered two consecutive generations only. We now develop a mathematical framework for large waiting times, that is, for a large number of generations (

The cascaded channel model. (a) The transition from one generation to the next can be modelled as communication of information through a noisy channel. Thus, long waiting times can be modelled as a cascade of similar channels. (b) The equivalent channel for

Since the matrix

We now consider the real-life scenario when the population is

The given theory can be easily extended to study the nonhierarchical cyclic interactions between

Another interesting aspect of cascaded channels is study of mutual information between consecutive generations. A discrete channel is a system with input alphabet

Extensive computer simulations were carried out to test the condition for biodiversity, as predicted by (

Figure

Typical plot of the eigenvalues of resultant probability matrix

Simulations were also carried out for different values of

The probability of extinction,

The proposed mathematical framework can be conveniently used to study systems with unequal reaction rates by appropriately modifying the transition matrix,

The eigenvalues of this generalized transition probability matrix can be used to study the effects of the different parameters on biodiversity. As an illustrative example, suppose we wish to investigate the effect of unequal (normalized) reproduction rates

Another interesting observation from nature is the fact that resource competitors can benefit one another through containment of shared competitors [

(a) Competitive network with

Figure

Mutual information,

We have developed a framework using cascaded channel model for analyzing the existence of biodiversity in nonhierarchical interactions among

The significant contributions of this paper are as follows.

The transition between successive generations is modelled as communication of information through a

The cascaded channel model exploits a very unique characteristic of matrices: the ability to express square matrices in terms of their eigenvalues. This approach permits a cascade of channels to be modelled as a product of several matrices. This approach leads to an elegant solution in terms of the power of the eigenvalues.

The versatility of this approach allows it to be extended to study more than three interacting species, species with ecosystems with unequal rates for different species and competitive networks with different competitive relationships.

The rate of transfer of information over successive generations is studied using mutual information, and it is found that “greedy” information transfer between successive generations may lead to conditions for extinction.

Explicit dependence of the probability of population distribution on the transition probabilities linked to the reproduction rate, selection rate, and mobility is derived in this paper. The condition for biodiversity is derived, which is corroborated by simulations.

This generalized mathematical framework can be used to study biodiversity in any number of interacting species, ecosystems with unequal rates for different species, and also competitive networks.

The author declares that there is no conflict of interests regarding the publication of this paper.