DOI: 10.5061/dryad.sf7m0cgc9
title: Seasonal density-dependence can select for partial migrants in migratory species

description: Whether, and which, individuals migrate or not is rapidly changing in many
 populations. Exactly how and why environmental change alters migration
 propensity is not well understood. We constructed density-dependent
 structured population models to explore conditions for the coexistence of
 migrants and residents. Our theoretical models were motivated by empirical
 data identified via a systematic literature review. We find that the
 equilibrium density in the season with the strongest density-dependence of
 a strategy predicts whether the strategy will become dominant within the
 population. This equilibrium density represents strategy fitness in a
 seasonal environment and can be used to examine selection on migratory
 behaviour. Whether partial migration can be maintained within a population
 depends on where in the annual cycle density-dependence operates.
 Diversified bet-hedging, where parents produce a mix of migrants and
 residents, also maintains partial migration. Our study disentangles
 density-dependent and density-independent rates in a population with
 seasonal structure, potentially providing routes to explain the rapid
 change in migration strategies observed in many populations.
description: We investigate how different density-dependent regimes influence
 the coexistence of migratory and resident strategies within a population
 by developing a theoretical model. Our structured population model is
 motivated by empirical evidence from a literature review. In our model of
 a partially migratory population, migrants and residents share breeding
 grounds and overwinter apart. Based on such seasonal population structure,
 we constructed two monomorphic models where only migrants or residents
 exist in the population to calculate their strategy-specific equilibrium
 density for each season, and a polymorphic model with two strategies
 competing in the same population to calculate the proportion of each
 strategy at equilibrium. We can thus explore when two strategies coexist
 and when they cannot by modelling different scenarios.
description: # Seasonal density-dependence can select for partial migrants in migratory
 species
 [https://doi.org/10.5061/dryad.sf7m0cgc9](https://doi.org/10.5061/dryad.sf7m0cgc9) In this study, we investigate how different density-dependent regimes influence the coexistence of migratory and resident strategies within a population by developing a theoretical model. Our structured population model is motivated by empirical evidence from a literature review. In our model of a partially migratory population, migrants and residents share breeding grounds and overwinter apart. Based on such seasonal population structure, we constructed two monomorphic models where only migrants or residents exist in the population to calculate their strategy-specific equilibrium density for each season, and a polymorphic model with two strategies competing in the same population to calculate the proportion of each strategy at equilibrium. We can thus explore when two strategies coexist and when they cannot by modelling different scenarios. We then examine how density-dependent reproduction, density-dependent winter survival, and density-independent impacts influence the population density in the breeding and non-breeding season for both the monomorphic and polymorphic models, and affect the proportion of migrants in the polymorphic model. ## Description of the data and file structure ### JL\_migration\_selection\_code250126.Rmd This file is the R script for our models. The definition of each parameter in the model is provided below. Parameters | Definitions ai | Density-independent rate in reproduction function for stage i. bi | Strength of density-dependence in reproduction function for stage i. bn | Strength of density-dependence in reproduction function for residents. bm | Strength of density-dependence in reproduction function for migrants. ci | Density-independent rate in winter survival function for stage i. di | Strength of density-dependence in winter survival function for stage i. dn | Strength of density-dependence in winter survival function for residents. dm | Strength of density-dependence in winter survival function for migrants. p | The probability of producing resident juveniles by resident adults. q | The probability of producing migratory juveniles by migratory adults. dd_bi | The density-dependent reproduction function for stage i, which is equivalent to E6 in the main text. dd_wsi | The density-dependent winter survival function for stage i, which is equivalent to E8 in the main text. fi | The default reproductive rate in the model for stage i. swi | The default winter survival rate in the model for stage i. sbi | The default survival rate in the breeding season for stage i, which is equivalent to S_(I,i) in the main text. miui | The default transmitting rate for stage i, which is equivalent to miu_i in the main text. A1 | The projection matrix for the whole year, after the non-breeding season, which is equivalent to A(N(t)) in the main text. A2 | The projection matrix for the whole year, after the breeding season. mat_Ds | The projection matrix for the breeding season, which is equivalent to matrix B_I in the main text. mat_Dw | The projection matrix for the non-breeding season, which is equivalent to matrix B_II in the main text. nt | The population vector for the breeding season, which is equivalent to N_I(t) in the main text. ntw | The population vector for the non-breeding season, which is equivalent to N_II(t) in the main text. kb_n, kw_n | Strategy-specific density of residents at equilibrium at the start of breeding or non-breeding season in the residents-only model, which is equivalent to k_(x,n) in the main text. kb_m, kw_m | Strategy-specific density of migrants at equilibrium at the start of breeding or non-breeding season in the migrants-only model, which is equivalent to k_(x,m) in the main text d_kb, d_kw | Difference between the strategy-specific densities of migrants and residents at the start of the breeding or non-breeding season, which is equivalent to Δk_x in the main text. Kb_n, Kw_n | Density of residents at equilibrium at the start of breeding or non-breeding season in the polymorphic model, which is equivalent to K_(x,n) in the main text. Kb_m, Kw_m | Density of migrants at equilibrium at the start of breeding or non-breeding season in the polymorphic model, which is equivalent to K_(x,m) in the main text. rm | The proportion of migrants at equilibrium in the polymorphic model, which is equivalent to R_m in the main text. ### model inputs and outputs.xlsx This file reports all parameter sets of each modelling scenario and according model outputs, which supplements Table S3 in Appendix S1. Parameters are consistent with those in Table S3 and Table 2. ### multiple\_equilibria.csv This file is the data to get the Fig.S3, which was produced by changing the initial population structure of each model. Variables | Descriptions stage.class | The stage class. n | The number of individuals in each stage class of the initial population state vector in the first year. Rm | The proportion of migrants in the population when the model reach equilibrium. ## Code/Software The model is coded in R.