Simulation of animal behavior patterns in context of evolutionary stable strategies.

Introduction

Evolutionary Stable Strategy (or ESS; also evolutionarily stable strategy): A behavioral strategy (phenotype) if adopted by all individuals in a population that cannot be replaced or invaded by a different strategy through natural selection.¹ In game-theoretical terms, an ESS is an equilibrium refinement of the Nash equilibrium, being a Nash equilibrium that is also "evolutionarily stable."

Rules

When two animals contest, the outcome is determined by their behavior type. Only one animal can get the prize, wasting time and getting injured are punished. The simplified point system² is as follows:

• Winning: 50 points
• Losing: 0 points
• Injury: -100 points
• Wasting Time: -10 points

For now let's consider only two behaviors, Dove and Hawk. The Dove will spend time contesting the prize non-violently (posing etc.) until its opponent gives up. If attacked immediately flees leaving the prize behind but avoiding injury. The Hawk will fight for the prize until victorious or injured. With these behaviors defined we can get the following table of expected average outcomes:

	Dove	Hawk
Dove	15	0
Hawk	50	-25

Value from each cell represents how much on average will a behavior from it's row get when confronted with another behavior from it's column. For example: A dove when contesting with another dove will on average earn 15 points each, because they will both waste time $(2 \times -10)$ and only one of them will get the prize $(+50)$, this gives average value of $\dfrac{2 \times -10 + 50}{2} = 15$.

Simulated population consisting of only doves and hawks will reach an equilibrium with the behavior ratio 5:7 (see Figure 1.1. and 1.2.). This population is stable, because if there were any less doves or hawks their expected reward would be higher than other behavior's and their percentage would increase (see Figure 1.3.). The same ratio can be achieved by solving a system of equations based on the outcome table with d and h variables and a parameter n:

$$ \begin{cases} 15 \times d + 0 \times h = n\\ 50 \times d - 15 \times h = n\\ \tag{1} \end{cases} $$

This means that neither hawks or doves will gain advantage over the other and their expected rewards are equal. We also don't assume, that an animal can act in only one way, so acting like a dove with probability $\dfrac{5}{12}$ and acting like a hawk with probability $\dfrac{7}{12}$ could be considered an Evolutionary Stable Strategy

import numpy as np
import matplotlib.pyplot as plt
import matplotlib
from simulation import PopulationAnalytical, expected_rewards
from plot_data import plot_population, plot_ratios, theme_init

%matplotlib inline
%config InlineBackend.figure_format = 'svg'

theme="light"
theme_init(theme)

outcome_matrix = np.array([[15, 0],
                           [50, -25]])

balanced_population = PopulationAnalytical(size=50000, generation_count=75,
                                           fitness_offspring_factor=0.1, random_offspring_factor=0.0,
                                           outcome_matrix=outcome_matrix,
                                           behaviors=(0, 1), starting_animal_ratios=(1, 1))

dove_dominant_population = PopulationAnalytical(size=50000, generation_count=75,
                                                fitness_offspring_factor=0.1, random_offspring_factor=0.0, 
                                                outcome_matrix=outcome_matrix, 
                                                behaviors=(0, 1), starting_animal_ratios=(10, 1))
balanced_population.run_simulation()
dove_dominant_population.run_simulation()
plot_population(balanced_population, "Fig. 1.1. Balanced dove and hawk population coming to an equlibrium in a simulation", theme=theme)
plot_population(dove_dominant_population, "Fig. 1.2. Dove dominated dove and hawk population coming to an equlibrium in a simulation", theme=theme)
plot_ratios(expected_rewards(outcome_matrix), description="Fig. 1.3. Expected rewards in dove and hawk population,\nequal at x = 5/12", theme=theme)

Rules expansion

Let's introduce another 3 behaviors:

• Retaliator: Will act like a dove until attacked, then it will retaliate.
• Bully: Will act like a hawk until getting attacked, then will act like a dove.
• Prober-Retaliator: Will act like a retaliator, but sometimes probes the contestant by attacking.

We will also use the advanced version of outcome table³, for it has been studied more and is more complex:

	Dove	Hawk	Retaliator	Bully	Prober-Retaliator
Dove	29	19.5	29	19.5	17.2
Hawk	80	-19.5	-18.1	74.6	-18.9
Retaliator	29	-22.3	29	57.1	23.1
Bully	80	4.9	11.9	41.5	11.2
Prober-Retaliator	56.7	-20.1	26.9	59.4	21.9

And from now on, instead of table we will use the outcome matrix with the ordering like in the table above:

$$ \left[\begin{array}{cc} 29 & 19.5 & 29 & 19.5 & 17.2\\ 80 & -19.5 & -18.1 & 74.6 & -18.9\\ 29 & -22.3 & 29 & 57.1 & 23.1\\ 80 & 4.9 & 11.9 & 41.5 & 11.2\\ 56.7 & -20.1 & 26.9 & 59.4 & 21.9\\ \end{array}\right] \tag{2} $$

outcome_matrix = np.array([[29, 19.5, 29, 19.5, 17.2],
                           [80, -19.5, -18.1, 74.6, -18.9],
                           [29, -22.3, 29, 57.1, 23.1],
                           [80, 4.9, 11.9, 41.5, 11.2],
                           [56.7, -20.1, 26.9, 59.4, 21.9]])

balanced_population = PopulationAnalytical(size=50000, generation_count=1000, 
                                           fitness_offspring_factor=0.1, random_offspring_factor=0.0, 
                                           outcome_matrix=outcome_matrix, 
                                           behaviors=(0, 1, 2, 3, 4), starting_animal_ratios=(1, 1, 1, 1, 1))
dove_dominant_population = PopulationAnalytical(size=50000, generation_count=500, 
                                                fitness_offspring_factor=0.1, random_offspring_factor=0.0, 
                                                outcome_matrix=outcome_matrix, 
                                                behaviors=(0, 1, 2, 3, 4), starting_animal_ratios=(36, 1, 1, 1, 1))
balanced_population.run_simulation()
dove_dominant_population.run_simulation()
plot_population(balanced_population, "Fig. 2.1. Population made up of all behaviors with equal starting percentages\nbeing dominated by probers in a simulation", theme=theme)
plot_population(dove_dominant_population, "Fig. 2.2. Dove dominant population made up of all behaviors with being\ndominated by an equilibrium of hawks and bullies in a simulation", theme=theme)

The population made up of all behaviors with equal starting percentages ends up being dominated by probers-retaliators and other behaviors die out (see Figure 2.1.). However if the starting population is different, for example dominated by doves, the final population can be dominated by an equlibrium of hawks and bullies⁴ (see Figure 2.2.). The outcome of the simulation may seem difficult to predict, but we will try to find an analytical method to predict whether a population composed of given behaviors can be evolutionarily stable.

Analytical approach

Assumptions and definitions

For our first analytical model we will assume the following:

• There are no random mutations, the "random offspring factor" parameter is set to 0.
• The population of any behavior cannot fall below 0.

Let's call the outcome matrix (1) $M$

We will use the given notation for contructing submatrices:

$$ M[a_{1}, a_{2}, \dots, a_{r}; b_{1}, b_{2}, \dots, b_{s}] = \left[\begin{array}{cc} m_{a_{1}, b_{1}} & m_{a_{1}, b_{2}} & \dots & m_{a_{1}, b_{s}}\\ m_{a_{2}, b_{1}} & m_{a_{2}, b_{2}} & \dots & m_{a_{2}, b_{s}} \\ \vdots & \vdots & \ddots & \vdots\\ m_{a_{r}, b_{1}} & m_{a_{r}, b_{2}} & \dots & m_{a_{r}, b_{s}} \\ \end{array}\right] \tag{3} $$

where ${ a_1, a_2, \dots, a_r }$ are the indices of the $r$ selected rows and ${ b_{1}, b_{2}, \dots, b_{s} }$ are the indices of the $s$ selected columns. For example let's construct matrix $M_{0}$ by taking elements from $M$ that are on rows 1, 4 and colums 1, 4:

$$ M_{0} = M[1, 4; 1, 4] = \left[\begin{array}{cc} 29 & 19.5 \\ 80 & 41.5 \\ \end{array}\right] \tag{4} $$

We will define population as a n-element vector, where n is the amount of behaviors in the population and it's elements are the amounts of population membert that have given behavior.

• Sum of the elements of population is constant and equal to initial population size.
• All elements are non-negative.

We will also define different types of stability:

• Weak stability: the population is stable with this exact behavior distribution, but there exists a vector $v$ with the same dimension as given population and $\forall\varepsilon\in\mathbb{R}_{+} ||v|| < \varepsilon$, that when added to the population will destabilize it.
• Strong stability: the population is stable and any vector $v$ with the same dimension as given population and $\forall\varepsilon\in\mathbb{R}_{+} ||v|| < \varepsilon$ when added to the population will cause it to return to the state before the addition.

Predicting the expected rewards

The necessary condition for a population consisting of behaviors ${a_1, a_2, \dots, a_r}$ to be strongly stable and necessary and sufficient to be weakly stable is that there exist $r$ positive variables ${x_1, x_2, \dots, x_r}$ and any parameter $c$ that satisfy a system of equations given by following matrix multiplication:

$$ M[a_1, a_2, \dots, a_r; a_1, a_2, \dots, a_r] \left[\begin{array}{cc} x_1 \\ x_2 \\ \vdots \\ x_r \end{array}\right] = \left[\begin{array}{cc} c \\ c \\ \vdots \\ c \end{array}\right] $$

When vector of variables that satisfy this system of equations is normalized, it's elements are proportions of corresponding behaviors in the result population.

Footnotes

1. Cowden, C. C. (2012). Game Theory, Evolutionary Stable Strategies and the Evolution of Biological Interactions. Nature Education Knowledge 3(10):6. ↩

2. Dawkins, R. (1989). The Selfish Gene (Anniversary edition). Oxford University Press. ↩

3. Maynard Smith, J., Price, G. (1973). The Logic of Animal Conflict. Nature 246, pp. 15–18. ↩

4. Gale, J., Eaves, L. (1975). Logic of animal conflict. Nature 254, pp. 463. ↩