Optimal Harvesting for a Predator-Prey Agent-Based Model using Difference Equations

Abstract

In this paper, a method known as Pareto optimization is applied in the solution of a multi-objective optimization problem. The system in question is an agent-based model (ABM) wherein global dynamics emerge from local interactions. A system of discrete mathematical equations is formulated in order to capture the dynamics of the ABM; while the original model is built up analytically from the rules of the model, the paper shows how minor changes to the ABM rule set can have a substantial effect on model dynamics. To address this issue, we introduce parameters into the equation model that track such changes. The equation model is amenable to mathematical theory—we show how stability analysis can be performed and validated using ABM data. We then reduce the equation model to a simpler version and implement changes to allow controls from the ABM to be tested using the equations. Cohen’s weighted $\kappa$ is proposed as a measure of similarity between the equation model and the ABM, particularly with respect to the optimization problem. The reduced equation model is used to solve a multi-objective optimization problem via a technique known as Pareto optimization, a heuristic evolutionary algorithm. Results show that the equation model is a good fit for ABM data; Pareto optimization provides a suite of solutions to the multi-objective optimization problem that can be implemented directly in the ABM.

Appendices

Appendix 1: Overview, Design concepts, and Details (ODD) Protocol for Rabbits and Grass

The model upon which this version is based is included in the sample library of NetLogo Wilensky (2009), a popular agent-based modeling platform. The description here is warranted as it includes the mechanics of an optimization problem, the details of which are not available elsewhere (including the NetLogo version). The following description follows the ODD protocol for describing agent-based models as proposed in Grimm et al. (2010).

1.1 Purpose

The purpose of this model is to examine population dynamics of a simple environmental system. In particular, it is a model of rabbits eating grass in a field. On each day of the simulation, poison can be placed on the field in order to kill the rabbits. The poison kills the rabbits with a certain efficacy but has no effect on the grass. The poison costs money, so there is some interest in minimizing the number of days on which poison is used. From this scenario, a natural multi-objective optimization problem arises: What poison schedule should be used in order to minimize the total number of rabbits alive during the course of a simulation while also minimizing the amount of poison? This version of the model is an attempt to answer this question.

1.2 Entities, State Variables, and Scales

This section contains a description of the grid cells, spatial and temporal scales, and the rabbits. It also contains a description of the format of a poison strategy, the investigation of which is the key feature of the model.

Grid cells, spatial scale, and temporal scale The world is a square grid of discrete cells, representing a field. The grid is toroidal, meaning that edges wrap around both in the horizontal and vertical directions. The distance from the center of a cell to a neighboring horizontal or vertical cell is 1 U (thus the distance between two diagonal cells is $\sqrt{2}$). Units are abstract spatial measurements. Time steps are also abstract discrete units. A simulation consists of a finite number of time steps. The only state variable for each cell indicates whether or not the cell currently contains grass. When grass is eaten on a grid cell, there is a certain probability that it will grow back at each time step. This growth happens spontaneously (Table 4).

Table 4 Grid cell state variables

Table 5 Rabbit state variables

Table 6 Poison schedule details

Rabbits Each time step, rabbits move, eat grass (or not), and reproduce (or not). Reproduction is asexual and based on energy level, which is raised when a rabbit eats. Rabbits lose energy both by moving and by spawning new rabbits. If a rabbit’s energy level drops to 0 or lower, the rabbit dies (Table 5).

Poison schedule A poison schedule $u$ is a vector of length $total\_sim\_time$, with each entry either 0 or 1. Each entry corresponds to one time step in the simulation; 0 means that poison is not used and 1 means that poison is used. Thus, there are a total of $2^{total\_sim\_time}$ possible poison schedules. The poison has a maximum strength which degrades over time with repeated use. If the poison is not used, the strength increases again, up to the maximum. Poison is not spatially dependent; hence the probability of a rabbit being poisoned does not depend upon the rabbit’s location in the grid (Table 6).

Poison strength $s$ is determined by the equation

$$\begin{aligned} s_(t\!+\!1)=(1\!-\!p_{deg}) \cdot u(t) \cdot s(t) \!+\! (1-u(t)) \cdot (s(t)+p_{deg}(p_{max} - s(t))), \qquad s(0) = p_{max}.\nonumber \\ \end{aligned}$$

(10)

1.3 Process Overview and Scheduling

In order to minimize ambiguity, the model process is presented here as pseudocode.

1.4 Design Concepts

Basic principles In essence, this model is a predator-prey system, wherein the rabbits are predators and the grass is prey. Introduction of poison into the model, and having that poison modeled as a direct external influence on population levels, creates a natural setting for an optimization problem. One can study the effect of various poison strategies on population levels—in terms of minimizing the rabbit population, it can be thought of as a harvesting problem, but in terms of minimizing poison, it can be thought of as resource allocation.

Emergence Rabbit population and grass levels tend to oscillate as the simulation progresses. The frequency and amplitude of these oscillations can be affected by parameter settings and initial values and hence may be described as emergent model dynamics.

Interaction Agent interaction is indirect: Since rabbit movement is executed serially, it is possible that other rabbits deplete all of the grass in a particular rabbit’s potential field of movement, thereby reducing or eliminating the chance for that rabbit to gain energy.

Stochasticity Rabbit movement is totally random in that they cannot sense whether neighboring grid cells contain grass or not. Whether grass grows back on an empty grid cell is also random and a grid cell that has been empty for several time steps is no more likely to grow grass than a cell which has only just become empty. Rabbit death due to poison is a stochastic process dependent on the strength of the poison for that day.

Observation Rabbit population and grass counts are recorded at each time step. The total number of rabbits alive during the course of a simulation serves as a measure of fitness of the poison schedule.

1.5 Initialization

At initialization, $20\,\%$ of the grid cells contain grass; these are chosen at random. There are 120 rabbits placed at random locations throughout the grid; each begins with a random amount of energy between 1 and 8 inclusive. Total simulation time is 100 time steps, and each simulation contains a poison schedule $u$, described in “Entities, State Variables, and Scales” of Appendix 1.

1.6 Input Data

The poison schedule may be considered input data to the model; however, there is no external input data.

1.7 Submodels

There are three types of rabbit movement investigated in this study. In the first, referred to as random jump, rabbits jump to a grid cell selected at random from the entire space. The second type of movement is wiggle, wherein rabbits execute the following commands: face left by a random amount up to $45^\circ$, face right by a random amount up to $45^\circ$, move forward 1 U. The latter is similar to rabbits selecting a direction from a uniformly random distribution centered at their current heading and thus is meant to represent a more realistic movement scheme. The final movement scheme is referred to as neighbor 8 movement: Under this rule, rabbits move to the center of one of the eight neighboring grid cells. In all movement schemes, the energy lost by moving is the same regardless of the distance moved. The body of the text makes clear which movement scheme is being considered at any given time. Under all movement schemes, a movement costs $move\_cost$ regardless of the distance moved. Each rabbit moves exactly once per time step.

1.8 Optimization

Since the multi-objective optimization problem is the key feature of the model as presented here, several clarifying details are in order. The objectives of the optimization problem are to determine, for the parameter values provided, a set of Pareto optimal poison schedules which minimize the number of rabbits while also minimizing the amount of poison used. The number of rabbits refers to the total number of rabbits alive during the course of a simulation, not just those alive at the end of the final time step. Since a poison schedule is a binary vector of length $total\_sim\_time$, the amount of poison used is represented by the sum of the entries of that vector.

Appendix 2: Difference Equation Models

Figures 7 and 8 contain the full equation models discussed in the body of the text, including initial conditions. See Table 1 for a description of the terms used in the equations and the section “Entities, State Variables, and Scales” of Appendix 1 for state variable values.

Fig. 7

The discrete model (using random jump movement), tracking rabbits at different energy levels

Fig. 8

The updated model with movement parameters $m_0, \ldots , m_8$

Fig. 9

Comparison of ABM data (averaged over 100 simulations) with DE data after parameter estimation for four randomly chosen training schedules

Appendix 3: Parameter Estimation with respect to Control

Parameter estimation was performed in Matlab using the lsqcurvefit routine for minimizing the sum-of-squares error (SSE). Since the equations are to be used to solve a multi-objective optimization problem, it is necessary that the parameters are chosen such that the system responds to controls in the same way the ABM responds to those same controls. The parameters were fit to the following data: Twenty values were chosen at random between 1 and 50; these represent the frequency values—for each frequency value $n$, five control schedules were generated randomly containing $n$ 1’s and $100 - n$ 0’s; this provides a stratified random sample of poison schedules. These 100 poison schedules were implemented in the ABM and subsequently used as training data for the difference equation parameters: The routine was set up to minimize the SSE over all of the training data. Figure 9 shows plots of the ABM versus DE data for four randomly chosen schedules from the training data.

Appendix 4: Optimal Poison Schedules

Figure 10 contains four schedules from the final generation of the Pareto optimization algorithm. Under each poison schedule is the number of poison days and the average number of rabbits alive when that schedule is applied. Note that in general, periods of poisoning are punctuated by periods where poison is not applied. This is likely in order for the poison efficacy to return to full (or near-full) effectiveness.

Fig. 10

Several Pareto optimal poison schedules (to be read from left to right a 10 poison days; result: 31.15 rabbits b 20 poison days; result: 20.19 rabbits c 30 poison days; result 14.27 rabbits d 49 poison days; results 11.30 rabbits