DOI: 10.6084/m9.figshare.5683762.v4
title: Supplementary information to "Eco-Evolutionary Dynamics of a Population with Randomly Switching Carrying Capacity"

description: <b>Abstract</b><br>Environmental variability greatly influences the eco-evolutionary dynamics of a population, i.e. it affects how its size and composition evolve. Here, we study a well-mixed population of finite and fluctuating size whose growth is limited by a randomly switching carrying capacity. This models the environmental fluctuations between states of resources abundance and scarcity. The population consists of two strains, one slightly faster than the other, competing under two scenarios: one in which competition is solely for resources, and one in which the slow (“cooperating”) strain produces a public good. We investigate how the coupling of demographic (internal) noise with environmental randomness (external noise) affects the populations eco-evolutionary dynamics. By analytical and computational means, we study the correlations between the population size and its composition, and discuss the “eco-evolutionary game” underpinning the cooperation dilemma of public good production in a fluctuating environment. We determine when producing a public good is the best strategy; when it is beneficial but outcompeted by the non-producing fast strain, and when its production it is detrimental for cooperators. We also analyze the joint effect of demographic and environmental noise on the population size distribution.<b><br></b><b>Contents</b><br>The text file contains supplementary information to <i>Eco-Evolutionary Dynamics of a Population with Randomly Switching Carrying Capacity</i> by K.Wienand, E. Frey, and M. Mobilia.<br><br>The data_and_processing.zip archive contains the simulation data and the <i>Mathematica</i> notebook to process it in order to produce results and figures used in the paper. Structured as follows:<br><i>Part 1: The PDMP Approximation</i>The first portion of the sheet is focused on the description of the system by means of a Piecewise-Deterministic Markov Process (PDMP).<br>A PDMP provides an accurate description of the effects of the switching carrying capacity on the population size. In this section we show that, using the PDMP stationary solution, it is possible to describe the joint distribution of population size and environmental state. Thanks to this result, we are also able to describe the environment-conditioned distribution size.<br>In the main text, we proposed an Ansatz to compute the fixation probability for the slower strain (cooperators) in the pure competition scenario. Specifically, we average the fixation probability from the fitness-dependent Moran model over the stationary solution of the PDMP. Here we present results that demonstrate the accuracy of these predictions, and that the quality of the approximation deteriorates as the timescales of size and composition become less separate.We also assess the accuracy of the predictions from the effective approach we propose to tackle the public good scenario. Finally, we apply analogous procedures to compute the mean fixation times.<br><i>Part 2: Eco-evolutionary Games</i>The higher the public good benefit, the lower the fixation probability of cooperators. However, when they do fixate, they reach higher population sizes. This setting can be framed as an eco-evolutionary game, in which the payoff for a strain is the expected number of individuals it will have in the population after fixation. The lower fixation probability for cooperators represents the cost they incur.<br>In this section, we analyze the main features of this eco-evolutionary game. First, we characterize the correlation between size and composition, which is at the root of the game. Specifically, we analyze how it depends on the public good parameter and on the environmental switching rate.<br>Then we analyze directly the "payoffs" of the game. We explore how the payoff for cooperators changes with the environmental switching rate, the public good parameter <i>b</i>, and the selection strength. With these results, we define the value <i>b*</i> which yields the maximum payoff, the value at which the payoff is the same for both strains, and the critical value <i>b<sub>c</sub></i> at which cooperation becomes entirely detrimental for cooperators. Using the PDMP approximation, we can predict the payoffs and the critical values for any number of parameters.<br><i>Part 3: The Linear Noise Approximation</i>The PDMP approximation, while sufficient for much of the analysis, completely disregards the effect of demographic fluctuation on the population size distribution. In this section we present the results of using a Linear Noise Approximation to include demographic fluctuations.<br>The archive source_code.zip contains the source code used for simulations (requires the Boost C++ library).