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Simulating Rodent Populations Using Neural Networks
Paul Garratt 1 and J. Dickman1
Abstract
Agricultural businesses must manage pests effectively. Populations of pests and other
economically important species e.g. microtine rodents can exhibit violent fluctuations whose
causes are poorly understood. The Periodic Lethal Toxin Production, PLTP, hypothesis of
Doncaster and Plesner-Jensen attempts to explain the precipitous decline phase of rodent
cycles. Indirect and anecdotal observations support the hypothesis that the rodents are
poisoned by plants. This explanation remains untested experimentally.
This paper reports our design, development and use of a simulator to test the PLTP
hypothesis in a virtual environment. The simulator incorporates six features important in the
natural environment of the rodent rodents: breeding and death of the individual animals and
their predators, growth of their food plants, selection of food plants by the rodents, predation
on the rodents, migration of the rodents and predators and evolution of toxin resistance in
individuals. A mathematical treatment of so many factors would be impossible. The simulator
models the behaviour of individual animals active in a grid of zones where changing densities
of rodents, plants and predators comprise the virtual environment. The individuals use a
neural network to make important decisions such as choice of food and migration routes
through the grid. The neural network responds to factors such as food availability and
predator threat. The behavioural response to the environment evolves by means of a genetic
algorithm applied to the virtual chromosomes of the individual rodents. The individuals grow
and breed at rates proportional to their food intake. Predation reduces the population to
provide a turnover of individuals within the system. The inclusion in the model of toxic plants
and toxin resistance allows testing of the PLTP hypothesis. An individual rodent can
spontaneously gain or lose resistance to plant toxins by mutation. An individual can inherit
resistance if either parent has it. Modelled plants produce toxins when over-grazed, as
observed in nature.
The discrete event simulator executed successfully exploring the dynamics of rodent
populations to test hypotheses which are pre-conditions to PLTP such as: do the rodents feed
on toxic plants when the density of preferred plants is below a critical thresh hold? The model
supported and refuted several similar hypotheses in the virtual environment. The simulation
exercise replicated the cyclical population changes observed in natural rodent populations.
For some cases involving migration across the zones in the system the neural network
appeared to give authentic results: the cyclical fluctuations in population density becoming
synchronised. However in more complex situations the model tended to a non-cyclic
equilibrium. The novel simulation design combining neural networks and genetic algorithms
proved to be very powerful. The model programmed in an extensible object-oriented manner
allowed simulation of a variety of situations. While results fell short of validating the PLTP
1
Department of Electronics and Computer Science, University of Southampton, UK.
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hypothesis in full, the use of a neural network to model decision-making by animals under
study holds great promise for future investigations.
1.
Introduction
In their paper “Lethal Toxins in Non-preferred Foods: How Plant Chemical
Defences Can Drive Microtine Cycles” (Journal of Theoretical Biology, July 1999),
Susanne Plesner-Jensen and C. Patrick Doncaster present a theory called “Periodic
Lethal Toxin Production” (PLTP). This theory outlines an explanation of how the
cyclical patterns in the size of some rodent populations could occur.
1.1 Rodent population dynamics
The reason behind fluctuations in rodent populations (e.g. voles and lemmings) is
an unanswered question in ecology (Kent 2001). The rodent population cycle lasts
between two and five years and progresses as follows (Krebs et al 1973):
1. Increase
2. Peak
3. Catastrophic decline
4. Extended low phase
Many theories as to the cause of the changes in population size have been
proposed but none has gained widespread acceptance. The four main mechanisms
that have been considered for the driving force behind the cycles are (Kent 2001):
Social organisation and dispersal
Hestbeck’s social fence hypothesis asserts that a dense population will inhibit
movement and dispersal, causing resources in the area to become depleted and the
population to crash.
Maternal effects
Rood & Boonstra have shown that if the density of female rodents is reduced then
young in the population have a higher survival rate and reproductive ability.
Therefore young born into a high density population are able to produce less
offspring in the next generation, causing a crash.
Predation
As the rodent population increases, so does that of the predator Cycling in this case
depends on the ability of the rodent population to recover faster than the predators.
Nutrition and food
Rodent populations are influenced by both the quality and quantity of the food they
consume.
The PLTP hypothesis was proposed by Plesner-Jensen and Doncaster. The
hypothesis is based on toxin production by plants but differs from earlier
explanations: the production is not constant but in direct response to grazing by the
rodents.
In the hypothesis, the population cycles are proposed to be a result of complex
interactions between four species within two distinct environments: tundra and the
surrounding area. The species are:
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 Rodent
 Predator, which feeds on the rodent.
 The preferred food of the rodent
 The non-preferred food of the rodent, having the ability to produce toxins.
The plant species both have a logistic growth pattern, modelling the concept of a
‘carrying capacity’, the maximum density that can be supported. A differential
equation
representing
the
growth
is:
ds
s

 r  s 1  
dt
 K
where s is the density of the plant, r is the growth rate and K is the carrying capacity.
An example with K=1 and r=0.2 is shown in figure 1.
Kent ran the model using parameters gathered from experimental and observational
data in existing literature on the subject. He was able to produce results that
corresponded to the observed data on the population cycles on five criteria. These
were:
 Cycle period
 Peak population density
 Trough population density
 Density ratio
 Occurrence of population collapse
In simulations without a toxic pulse only the first four of these could be
satisfied. When the toxic pulse was introduced, the model fulfilled all five criteria.
An example of a single cycle with the toxic pulse is shown in figure 2.
Density (ha-1)
250
200
150
100
50
0
0
0.5
1
1.5
2
2.5
Time (years)
Figure 2: Population density of rodents in tundra environment
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The effect of migration to and from surrounding areas is absent, as is the species
which preys on the rodents. The model does not cater for evolutionary factors, such
as the development of a resistance to the toxin by the rodent
2. System Design
The simulator described in this paper incorporates aspects of the situation that Kent
did not include.
The simulator models a number of factors that are present in the natural system:
 Breeding and death of rodents and predators
 Growth of plant species
 Selection of food by rodents
 Predation on rodent population
 Migration of rodents and predators between the regions
 Evolution of resistance to plant toxins
Rodent density
w1
Predator
density
Preferred plant
density
w2
Σ
w3
Response
w4
Non-preferred
plant density
Figure 3 Neural network determining rodent behaviour
The system is based on a grid structure, with each of the squares representing an
area of land in which the plants and animals would exist. Each zone keeps track of
the density of the plants and the number of rodents and predators within it. The
entities in the system, the rodents and the predators, have the freedom to move to an
adjacent square on each iteration of the simulation using a connectionist model
(neusciences 2003). For each choice, a set of stimuli indicates the density of the four
species in that direction. A weighted sum over nearby grid squares is used to
determine the level of a stimulus in a given direction see figure 3.
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2.1 Resistance to Toxin
Another aspect included in the model is the evolution of a resistance to the toxin by
the rodents. This is necessary to test Plesner-Jensen and Doncaster’s proposition that
resistance would not occur, as the population regrowth in areas where the toxic plant
is found is largely due to immigration from the surrounding regions. If both parents
of the new individual have resistance to the toxin then the offspring will also have
resistance. If neither parent has resistance then neither will the offspring. After the
initial resistance has been calculated then the effect of mutation is applied.
2.2 Implementation
The simulator had to be able to run a simulation of hundreds of years in a
reasonable time. To keep the implementation modular and discrete it is constructed
in an object-oriented language: C++.
Simulat
or
Plant
Animal
1
*
1
1
1
Preferre
d Plant
Predato
r
Zone
1
*
1
1
ToxicPl
ant
*
Rodent
Figure 4 The simulator class structure
2.1.1 Simulator structure
The root class in the system is the Simulator class. The Simulator controls the
sequence of events that occurs in the system and outputs the current state of the
simulation on each iteration. It holds references to each of the Zones in a two
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dimensional array. The Zone class controls the interactions between the different
species within the system. It has three main functions that are called from Simulator:
 stepSimulation
 moveAnimals
 finishMovement
2.1.1.1 step Simulation
This function calculates the interaction between the species within the zone.
The grow function of the plant species is called, increasing the density of plants
within the area. Next, the level of consumption of each of the varieties of plant by
the rodents is calculated. The same calculation is performed for the interaction
between the rodents and the predators. The decrease in density for the rodents is
added to the decrease due to predation. The numbers of rodents and predators that
are born are calculated from the increase in their densities. The animals that die are
also chosen randomly from the populations.
2.1.1.2 move Animals
The result of this function call corresponds to the direction in which the animal
wants to move.
2.1.1.3 Finish Movement
The purpose of this function is to move the animals from the list of incoming
predators and rodents to the main list of animals in the zone.
2.1.1.3 Animal
Animal contains the code that evaluates the desirability of moving in a given
direction. Its constructor takes pointers to two parent animals and contains the
genetic algorithm that evolves the weights of the neural network used for the
evaluation (Obitko 2002).
3.
Program Testing
The behaviour of the system as a whole is difficult to test due to the complex nature
of the interactions between the species. However, many aspects of the system could
be tested in isolation using small test harnesses developed for the purpose.
Test 1.1: Do the plants grow as expected when parameters from Kent’s simulation
are used? Outcome: PASS – the system simulated the expected logistic growth of
the plants
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Test 1.2: Does the rate of plant growth increase when the growth rate parameter is
increased? Outcome: PASS – the system simulated an increased rate of growth of
the plants.
Test 1.3: Is the density of the plants zero when the carrying capacity is zero?
Outcome: FAIL – a division by zero caused the program to crash. The bug was fixed
by adding a test for zero carrying capacity. The test was then passed.
3.1
Testing of animal behaviour within a zone
Test 2.1: Do the rodents only feed on the preferred food when the density of
preferred plants is above the critical threshold? Outcome: PASS – the rodents did
not eat any of the toxic plants
Test 2.2: Do the rodents feed on the toxic food when the density of preferred plants
is below the critical threshold? Outcome: PASS – the rodents did eat some of the
toxic plants.
Test 2.3: Does the mortality rate of the rodents increase when the density of the
toxic plants is low enough to cause toxin production? Outcome: PASS – the
mortality rate of the rodents did increase.
Testing of animal migration between zones
Test 3.1: Does the algorithm for calculating the stimuli perform as expected?
Outcome: FAIL– the stimuli to the south and west was also 2.17. The bug was due
to not raising the rate of weight decrease to the power of the distance travelled. A
subsequent retest showed the expected result.
Test 3.2: Does a rodent move from one zone to an adjacent zone as expected?
Outcome: PASS – the rodent moved as expected.
3.2 Reproduction of Kent’s results
The first simulation was to check that results matched those presented by Kent.
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100
90
80
Density (kg/ha)
70
60
Toxic plants
50
Preferred Plants
Voles
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Time (years)
Figure 5: Output of simulator with no toxic pulse
As expected, this output looks identical to Kent’s results for the population
dynamics without the toxic pulse see figure 5. Next, the toxic pulse was added and
again the results were compared with Kent’s. The toxic pulse was added by setting
the toxic threshold to 20, the value used by Kent. The procedure used was the same
as
before.
The
results
are
shown
in
figure
6.
100
90
80
Density (kg/ha)
70
60
Toxic Plants
50
Preferred Plants
Voles
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Time (years)
Figure 6: Output of simulator with toxic pulse
This is again the same result that was obtained by Kent. To check on the
progression of the evolution the data was plotted from the beginning of the
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simulation. The density of rodents within a single zone in the simulation is shown in
figure 7.
16
14
12
Density (kg/ha)
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Time (years)
Figure 7: Evolution of weights in the neural network leading to regular cycles in the
simulation
It is clear that the evolutionary strategy is working – at the beginning the migration
is random, causing large fluctuations in the number of rodents within the square.
Later, as the simulation progresses, the migration becomes more ordered, with the
cycles in all of the squares becoming synchronised, as can be seen from the densities
of the toxic plants in three of the zones in figure 8.
Figure 8: Synchronisation of cycles occuring in the simulator
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It is clear that the evolutionary strategy is working – at the beginning the migration
is random, causing large fluctuations in the number of rodents within the square.
Later, as the simulation progresses, the migration becomes more ordered, with the
cycles in all of the squares becoming synchronised, as can be seen from the densities
of the toxic plants in three of the zones in figure 8.
Figure 8: Synchronisation of cycles occuring in the simulator
The next step was to set up the simulation to include both the tundra region, where
the non-preferred plants grow, and the surrounding regions, where the toxic plants
are absent. This was done by increasing the grid size to 5 and setting the tundra
radius to 2, giving a circular region in the middle of the grid where the non-preferred
plants can grow. The results from this simulation were not as expected. The
simulation very rapidly became non-cyclic across the entire grid. This was very
disappointing, as we expected to see the cycling behaviour that had occurred in the
previous simulations, with migration in and out of the tundra reducing but not
halting the fluctuations. After this result, it was clear that any hope of producing the
population dynamics described by the PLTP hypothesis required considerable
extension of the simulator.
4.
Conclusions
The population of rodents crashes as a
predators into the surrounding regions.
rodents found there to suffer a similar
scenarios were produced. In the simpler
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result of the toxin, causing an exodus of
This subsequently causes the number of
decline. Results from simulating various
cases, the outcome was the same as Kent
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has achieved using his model. For some cases involving migration across the areas
in the system, the neural network approach seemed to work well: the cyclic
fluctuations in population density became synchronized. However, for the most
complex situations, the model tended to a non-cyclic equilibrium. The evolution of a
neural network to model the decision making by the animals was by far the most
novel part of the research, it was successful and holds the most promise for future
investigation.
Bibliography
Krebs, C.J., M.S. Gaines, B.L. Keller, J.H. Myers, and R.H. Tamarin (1973).
Population cycles in small rodents, Science 179, 35-41.
Plesner-Jensen and C. Patrick Doncaster (1999). Lethal Toxins in Non-preferred
Foods: How Plant Chemical Defences Can Drive Rodent Cycles, J. Theor. Biol.
199, 63-85.
Kent A. (2001). Rodent Cycles and the PLTP Hypothesis, University of
Southampton Faculty of Biomedical Sciences PhD thesis
Křivan V. (1996) Optimal foraging and predator-prey dynamics. Theor. Pop. Biol.
49, 265-290
Křivan V. & Asim Sikder (1999) Optimal foraging and predator-prey dynamics, II.
Theor. Pop. Biol. 55, 111-126.
Neusciences, http://www.neusciences.com (12 March 03)
Obitko M., Introduction to Genetic Algorithms. http://cs.felk.cvut.cz/~xobitko/ga/
(13 May 2002).
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