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Coevolutionary dynamics in one-to-many mutualistic systems

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Abstract

“One-to-many” mutualisms are often observed in nature. In this type of mutualism, each host individual can interact with many symbionts, whereas each individual symbiont can interact with only one host individual. Partner choice by the host is a potentially critical mechanism for maintaining such systems; however, its long-term effects on the coevolution between the hosts and symbionts have not been completely explored. In this study, I developed a simple mathematical model to describe the coevolutionary dynamics between hosts and symbionts in a one-to-many mutualism. I assumed that each host chooses a constant number of symbionts from a potential symbiont population, a fraction of which are chosen through preferential choice on the basis of the cooperativeness of the symbionts and the rest are chosen randomly. Using numerical calculations, I found that mutualism is maintained when the preferential choice is not very costly and the mutation rate of symbionts is large. I also found that symbionts that receive benefits from hosts without a return (cheater symbionts) and hosts that do not engage in preferential partner choice (indiscriminator hosts) can coexist with mutualist symbionts and discriminator hosts, respectively. The parameter domain of pure mutualism, i.e., free from cheater symbionts and indiscriminator hosts, can be narrower than the whole domain where the mutualism persists.

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Acknowledgments

This work was supported by the Grant-in-Aid for Scientific Research (C) from the Japan Society for the Promotion of Science (JSPS) KAKENHI 23570034.

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Correspondence to Hideo Ezoe.

Appendix

Appendix

The cooperativeness of symbionts was set to x i  = i/40 (i = 0, 1, …, 39). The symbiont choice trait of hosts was two-dimensional, where j = (j 1, j 2) and (c j , k j ) = (C j 1/40, K j 2/40) (j 1, j 2 = 0, 1, …, 39). The mutation rates of symbionts and hosts were

$$ {\mu}_{i{i}^{\prime }}=\left\{\begin{array}{cc}\hfill {\mu}_0/2\hfill & \hfill \mathrm{if}\kern0.5em \left|i-{i}^{\prime}\right|=1\kern0.5em \mathrm{and}\kern0.5em i\ne 0\hfill \\ {}\hfill {\mu}_0\hfill & \hfill \mathrm{if}\kern0.5em i=0\kern0.5em \mathrm{and}\kern0.5em {i}^{\prime }=1\hfill \\ {}\hfill 0\hfill & \hfill \mathrm{otherwise}\hfill \end{array}\right. $$
$$ {\nu}_{\left({j}_1,\;{j}_2\right)\left({j_1}^{\prime },\;{j_2}^{\prime}\right)}=\left\{\begin{array}{cc}\hfill {\nu}_0/4\hfill & \hfill \mathrm{if}\kern0.5em \left|{j}_1-{j_1}^{\prime}\right|+\left|{j}_2-{j_2}^{\prime}\right|=1\kern0.5em \mathrm{and}\kern0.5em {j}_1{j}_2\ne 0\hfill \\ {}\hfill {\nu}_0/2\hfill & \hfill \mathrm{if}\kern0.5em {j}_1={j_1}^{\prime },\kern0.5em {j}_2=0\kern0.5em \mathrm{and}\kern0.5em {j_2}^{\prime }=1\hfill \\ {}\hfill {\nu}_0/2\hfill & \hfill \mathrm{if}\kern0.5em {j}_2={j_2}^{\prime },\kern0.5em {j}_1=0\kern0.5em \mathrm{and}\kern0.5em {j_1}^{\prime }=1\hfill \\ {}\hfill 0\hfill & \hfill \mathrm{otherwise}\hfill \end{array}\right. $$

respectively. Throughout the numerical calculations, the value of ν 0 was 0.0001, and the turnover rate of the host population was α = 1.

Each numerical calculation began from the initial distributions of symbionts and hosts, where s i  = 0.025 (i = 0, 1, …, 39) and

$$ {h}_j=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill j=\left(0,0\right)\hfill \\ {}\hfill 0\hfill & \hfill \mathrm{otherwise}\hfill \end{array}\right. $$

respectively. Preliminary calculations showed that the initial distributions scarcely affected the final states of dynamics. Two million time units were sufficient for the dynamics to converge to a stable equilibrium or periodic oscillation.

In the final stages of the calculations, frequencies of the symbiont individuals of any trait value were not exactly equal to zero because of mutation. I determined that mutualist symbionts persisted when the frequency of cheater symbionts q = s 0s i was smaller than 0.995, whereas the cheaters were extinct when it was less than 0.005; cheaters and mutualists coexisted otherwise. Similarly, indiscriminator hosts became extinct when their frequency was less than 0.005.

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Ezoe, H. Coevolutionary dynamics in one-to-many mutualistic systems. Theor Ecol 9, 381–388 (2016). https://doi.org/10.1007/s12080-016-0296-x

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