# 2018 Annual Meeting of the Society for Mathematical Biology & the Japanese Society for Mathematical Biology

8-12 July 2018
Australia/Sydney timezone

## The coexistence and its sustainability in Batesian mimicry

11 Jul 2018, 16:40
20m
New Law School/--022 (University of Sydney)

### New Law School/--022

#### University of Sydney

60
Oral Presentation Ecology

### Speaker

Hayato Kato (Hokkaido University)

### Description

Batesian mimicry is a common phenomenon in nature, and it has been reported in various taxa. In Batesian mimicry, there are two species that have a similar colouration. One species is toxic or unpalatable, we call it “model-species”. The other is nontoxic or palatable, we call it “mimic”.

While many mathematical models focused on the evolution of mimicry, only a few mathematical models focused on community dynamics. Furthermore, the sustainability of the coexistence has not been studied. Here, we addressed the community dynamics and its sustainability in this study.

A part of the previous mathematical models has a common framework based on simple and plausible assumptions such as density-dependent effect and frequency-dependent predation. However, even a simple mathematical model based on the framework has not been analyzed completely. The simple mathematical model is also important mathematically because it is one of the basic models that have both density- and frequency-dependent effects and actually it includes a ratio-dependent predator-prey model.

The objective of our study is as follows; first, to solve a mathematical model based on the common framework completely and understand its properties. Secondly, to examine the impact of mimicry on community dynamics. Thirdly, to discuss the sustainable coexistence of model-species and mimic.

We focused on the case that the intrinsic growth rate of model-species is less than mimic because model-species pay some cost for acquiring toxicity and defined “predation impact (PI)” as the predation rate divided by the intrinsic growth rate. The result is divided into two types using PIs of model-species and mimic species; type I is when the predation impact of model-species is small, type II is when the predation impact of model-species is large. In type I, model-species and mimic always coexist. In type II, they can coexist, but not always. In particular, in type II, when the ratio of the carrying capacity change and become less than a threshold, model-species become extinct. Thus, the coexistence of model-species and mimic is unlikely to be maintained in type II under a varying environment. Therefore, Batesian mimicry in nature is supposed to be maintained in type I.

In our mathematical model, there are only three stable states; coexistence, mimic alone and both extinction. This result can explain the geographic distribution of model-species and mimic in more than 10 mimicry systems. For example, eastern coral snake is a toxic model and scarlet kingsnake is a nontoxic mimic in North America. In their geographic distribution, the areas of coexistence, mimic alone and both-absence were observed with latitude gradient. This pattern of the geographic distribution is consistent with the result of our mathematical model.

In addition to the above study, we construct some models, taking predator’s learning process into account. We present the results derived from these models and compare them.

### Primary author

Hayato Kato (Hokkaido University)