Evolutionary Dynamics of Fearfulness and Boldness: A Stochastic Simulation Model

A stochastic simulation model is investigated for the evolution of anti-predator behavior in birds. The main goal is to reveal the effects of population size, predation threats, and energy lost per escape on the evolutionary dynamics of fearfulness and boldness. Two pure strategies, fearfulness and boldness, are assumed to have different responses for the predator attacks and nonlethal disturbance. On the other hand, the co-existence mechanism of fearfulness and boldness is also considered. For the effects of total population size, predation threats, and energy lost per escape, our main results show that: (i) the fearful (bold) individuals will be favored in a small (large) population, i.e. in a small (large) population, the fearfulness (boldness) can be considered to be an ESS; (ii) in a population with moderate size, fearfulness would be favored under moderate predator attacks; and (iii) although the total population size is the most important factor for the evolutionary dynamics of both fearful and bold individuals, the small energy lost per escape enables the fearful individuals to have the ability to win the advantage even in a relatively large population. Finally, we show also that the co-existence of fearful and bold individuals is possible when the competitive interactions between individuals are introduced.

and the other boldness (detailed descriptions can be found in the main text). For our model, the other variables and scales are given below: (i) The population is composed of two pure strategists, R f -and R b -individuals. The offspring will have the same phenotype with their mother. The initial numbers of them are denoted as N f and N b .
(ii) All individuals are assumed to have the same maximum natural lifespan (or maximum survival age), denoted by T year old. The individual's maturity age for reproduction is one year old.
(iii) During a breeding season, the number of real predatory attacks is assumed to be a constant, denoted by a, and, similarly, the number of simple disturbing events is denoted by d.
(iv) The relative probability that a R f -individual is selected by the predators, compared with a R b -individual is denote as α ∈ (0, 1) . We also use β f to denote the probability that a R f -individual is captured when selected by the predator, and β b the probability that a R b -individual is captured when selected.
(v) During a breeding season, each individual would gain a certain amount of energy (E), and each escape would consume the energy of ε.
(c) Process overview and scheduling. According to the variable definitions and assumptions in (b) (see also the main text), the corresponding stochastic simulation (see also Section 2) is conducted: (i) At the starting of the process, both the R f -and R b -individuals are set averagely among different ages, which make up a population.
(ii) During each breeding season, the population would experience the predator attacks and disturbances. The specific probability that a R f -or R b -individual is killed by the predators is shown in the main text (see assumption (iii)).
(iii) The reproduction is assumed only occurs at the end of each breeding season. The total number of offspring born in a breeding season is exactly equal to the total number of the dead individuals due to the predator attacks and the limitation of individual's lifespan, according to the assumption that the total population size is assumed to be kept constantly at the end of each breeding season.
(iv) The numbers of R f -and R b -individual among the new born individuals are shown in the main text (see assumptions (iv) and (v)).
For given the initial condition (i.e. the initial proportions of R f -and R b -individuals), we run the simulation until the population becomes a pure strategy population (i.e. R fpopulation, or R b -population). We repeat this process 1000 times and then count the times that the R f -population occurs, denoted by C f , or the frequency that the R f -population occurs, denoted by p f = C f /1000.

(d) Design concepts.
(i) Emergence. The model was designed to explore the evolution of animal personality, i.e. fearful or bold. Population dynamics emerge from different personalities of the individual, and from the environment, i.e. attacks and disturbances.
(ii) Fitness. Fitness is measured automatically at the end of each breeding season, which directly determine the number of newly born R f -and R b -offspring.
(iii) Sensing. During each attack, the predator is assumed to know, without error, the personality of each individual in the population. Furthermore, the predators are assumed to be able to randomly sample from the population.
(iv) Interaction. The competitive interaction between fearful and bold individual is considered in the submodel (ii).   (i) In order to explore the effect of risk sharing on the evolution of fearfulness and boldness, we consider the situation where it is assumed that the number of newly born R f -and R b -individuals are independent of the numbers of the alive R f -and R b -individuals at the end of each breeding season, or that the probability that the fearful individuals (or bold individuals) are selected by the predators is independent of the population structure (i.e. it is frequency-independent). The more detail about this model can be found in the main text.
(ii) In order to explore the potential effect of competitive interactions between R f -and R b -individual, we also do simulations embody the frequency-dependent background fitness (see eq. 7 in the main text), and thus revise the corresponding numbers of the newly born R f -and R b -offspring according to the eq. 8 (see in the main text).

Stochastic simulation program in Matlab
In this section, our stochastic simulation program in Matlab is given below.