Calculating biodiversity under stochastic evolutionary dynamics

Highlights

• Extinction time is proposed as a new measure to study biodiversity driven under stochastic evolutionary dynamics.

• The state of biodiversity in stochastic dynamics differs from the deterministic dynamics, even under strong selection limit which mirrors the deterministic dynamics.

• The robustness of our results is proved under weak selection limit by forward equations.

• Our results highlight the importance of transient dynamics on biodiversity, which has been neglected in the deterministic dynamics.

Abstract

How biodiversity is maintained is one of the most important problems in biology. It, seemingly, challenges the Darwinian evolutionary theory which assumes that only the fittest species survive. Most previous researches try to solve the puzzle based on deterministic dynamics. However, it is not clear how demographic stochasity plays a role in biodiversity, even though randomness cannot be ignored in real biological systems. In contrast with deterministic methods, we propose that biodiversity is measured by the time that the species coexist. Modeled by the cyclic Rock-Paper-Scissors game, we concentrate on the Fermi process, where randomness is captured by the selection intensity. We find that there is a non-monotonic relationship between the selection intensity and the time that the species coexist. Intrinsic differences between stochastic dynamics and deterministic dynamics are shown to be present, even if the selection is in its strong limit which mirrors the deterministic dynamics. Furthermore, we prove that these results are robust for general stochastic imitation processes beyond the Fermi process. Our work highlights the importance of transient dynamics on biodiversity, which is absent in the deterministic counterpart, and opens an avenue to calculate biodiversity under stochastic evolutionary process.

Introduction

Biodiversity refers to the variety of living species on the earth. It is a challenge to the Darwinian evolutionary theory which assumes that only the fittest species survive. How biodiversity is maintained is a compulsive puzzle for biologists [1]. Previous researches try to solve the puzzle based on the Rock-Paper-Scissors game [2], [3], [4], where rock crushes scissors, paper wraps rock, and scissors cut paper. The Rock-Paper-Scissors game is not only a game for children, but also reveals the cyclic dominance phenomena which prevails ranging from real biological systems [2], [5], [6], human society [7], [8], [9] to economical systems [10]. And the cyclic dominance is likely to evolve through the assembly of unrelated types [11]. The side-blotched lizards [5] are a three-strategy cyclic dominance mating system, in which three distinct mating strategies are adopted by three kinds of males. The males are divided into three phenotypes according to the color of their throats: orange, blue and yellow. Orange-throated males are more advantageous than blue-throated males because they hold larger territories and have more chances (more females) for reproduction. Yellow-throated males are more advantageous than orange-throated males, because they sneak on females in the territories of orange-throated males. Yet yellow-throated males have no chance to invade the territories of the monogamous blue-throated males, so the blue thoated males are more advantageous than the yellow-throated males. The three kinds of male lizards are cyclic-dominant, which can be well captured by the Rock-Paper-Scissors game. Rock-Paper-Scissors game is thus a metaphor to study the biodiversity in such complex systems [2], [3], [4]. Typically, the payoff matrix of Rock-Paper-Scissors is given by:

If an individual adopts Rock as its strategy, and plays the game with another individual adopting Rock, it gets 0 as its payoff. If it plays the game with an individual adopting Paper, it pays $s(s>0)$ as its punishment. If it plays the game with an individual adopting Scissors, it wins and gets 1 as its reward. The payoff of an individual is not only dependent on its own strategy, but also is determined by the individuals it interacts with. Evolutionary game theory is a powerful tool to depict the evolution of such cyclic-dominant population, and fruitful results have been obtained [2], [4], [12], [13].

It has been assumed that the evolutionary dynamics is deterministic and the replicator equation is generally adopted to study this issue [2], [12], [14], [15]. Therein the stability of the population system is taken to capture whether the biodiversity exists. However, less attention is paid to the stochastic population (for exceptions, see [16], [17], [18], [19]). There are two kinds of stochasity in the real biological world. One is external. For example, the stochasity arising from the environment has a strong impact on bacterial growth [19]. And the environmental noise leads to the stable state of both linear and nonlinear dynamics [20]. The other is intrinsic stochasity. It is induced by the evolutionary process itself, such as mutation [20], [21], and the demographic stochasity induced by population fluctuations [22]. Selection intensity is an intrinsic noise of evolution. It depicts the strength of natural selection which is the driving force of evolution. It mirrors the inverse temperature in statistical physics. The stronger the selection intensity is, the bigger the evolution of the population is influenced by the cyclic-dominant games, and the less the randomness plays a role in the evolution. However, it is not clear how biodiversity is influenced by the selection intensity, which is one of the most important noises in evolutionary dynamics.

In the stochastic selection dynamics, the loss of biodiversity is sure to happen. Therefore, it is a matter of time to measure biodiversity. Here, it refers to how long the biodiversity is present. However, the time of the coexistence of species has seldom been systematically analyzed. Here we calculate the time that the three strategies coexist. And we are aiming to study the relationship between biodiversity and randomness in terms of selection intensity.

We focus on how the fixation time alters with the selection intensity. We find non-trivial results on fixation time. Furthermore, we make use of the stochastic transient dynamics to shed intuitive lights on these non-trivial results. Finally, we give a first step to analyse the time with the aid of partial differential equations.