1
Species Diversity and Predation Strategies in a
Multiple Species Predator-Prey model
Rory Mullan, David H. Glass and Mark McCartney
School of Computing and Mathematics, University of Ulster, Jordanstown
campus, Shore Road, Newtownabbey, Co. Antrim, BT37 0QB
mullan-r8@email.ulster.ac.uk, dh.glass@ulster.ac.uk,
m.mccartney@ulster.ac.uk
Abstract—A single predator, single prey ecological model, in which the behaviour of the populations relies upon two control parameters has
been expanded to allow for multiple predators and prey to occupy the ecosystem. The diversity of the ecosystem that develops as the model
runs is analysed by assessing how many predator or prey species survive. Predation strategies that dictate how the predators distribute their
efforts across the prey are introduced in this multiple species model. The paper analyses various predation strategies and highlights their effect
on the survival of the predators and prey species.
I.
INTRODUCTION
It is observable in nature that predator and prey population dynamics are directly reliant upon each other. The predator relies on
the prey to feed, and therefore survive, and the prey population is adversely affected by the success of the predators. Predatorprey modelling uses mathematical models to simulate real world ecosystems. These simulations calculate the effect that a
predator and a prey have on each other’s populations over time [1].
Predator-prey models can take the form of either a two species model, where a single predator predates upon a single prey, or a
multiple species model, where both multiple predators and multiple prey occupy the simulated ecosystem. Both forms can be
simulated using continuous time differential equations or discrete time difference equations [2]. The earliest predator-prey model
is the Lotka-Volterra model which consists of coupled differential equations originally described by Volterra [3] to describe the
interaction between predator-prey species and then independently arrived at by Lotka [4] to describe a chemical reaction. This
paper employs a multiple species implementation of a discrete time predator-prey model.
Within the field of ecological modelling much recent research has been undertaken looking at two species predator-prey models,
which have been used to investigate the underlying chaotic population dynamics [2, 5-9], the effect of the prey growth rate [10]
and to investigate population dispersal [11-13]. In their multiple species form, some work has been carried out investigating the
chaotic population dynamics in multiple species continuous time models [14-16]. Little research, however, has been undertaken
examining discrete predator-prey models in their multiple species form. Due to the increased computational power in computing
it is now possible to investigate the diverse ecosystems that develop employing discrete time multiple species predator-prey
models. Outside of ecological modelling, predator-prey models are also used for various other applications. For example, they
are used within economics [17,18] and in medical applications to model the immune system and its response to viruses or
tumours [19,20].
This paper uses a generalised multiple species form of a discrete time predator-prey model proposed by Neubert et al [9].
Predation strategies, which dictate how the predators go about hunting the prey, have been included in this model. The focus of
the analysis is on the final diversity (number of surviving predators and prey species) of the ecosystem that develops.
The two species discrete time predator-prey model will be introduced and generalised to allow for multiple predators and prey to
occupy its ecosystem. A discussion will then take place on how this model has been implemented. A simple predation strategy
that was initially used in this model is introduced, with a discussion then taking place on the ecosystems that develop utilising it.
Six further predation strategies are then introduced, along with the ecosystems that develop when they are used. The results from
the various predation strategies are then compared.
2
II.
MODEL
The following single species discrete time predator-prey model was proposed by Neubert et al [11]
𝑁𝑡+1 = 𝑁𝑡 𝑒 𝑟(1−𝑁𝑡−𝑃𝑡)
𝑃𝑡+1 = 𝑐𝑁𝑡 𝑃𝑡
(1a)
(1a)
(1b)
(1b)
where Nt represents the prey population at time step t and Pt represents the predator population at time step t. Two control
parameters here define the behaviour of the predator and prey: c defines the effectiveness of the predator at predating upon the
prey and r defines the reproduction rate of the prey. The Ricker model is being used for the behaviour of the prey.
This model is generalised to allow for multiple predators and prey to occupy the ecosystem. A direct extension of (1a) and (1b)
can written as
(𝑗)
(𝑖)
(𝑗)
(𝑗)
(𝑖𝑗)
𝑁𝑡+1 = exp(−(∑𝑚
𝑐𝑖𝑗 ) 𝑃𝑡 )) 𝑁𝑡 exp(𝑟𝑗 (1 − 𝑁𝑡 ))
𝑖=1(𝑓
(𝑖)
𝑃𝑡+1
(𝑗)
=
∑𝑛𝑗=1(𝑓 (𝑖𝑗) 𝑐𝑖𝑗 )𝑁𝑡(𝑗) 𝑃𝑡(𝑖)
(2a)
(2b)
(𝑖)
Where 𝑁𝑡 represents the jth prey species at time step t and 𝑃𝑡 represents the ith predator species at time step t , with cij and
𝑟𝑗 acting as the control parameters. Whereas (1b) can be generalised straightforwardly to (2b), (1a) is harder to generalise. A
direct generalisation of (1a) does not distinguish between the differing effectiveness of predator i in depleting the prey j’s
population as it contains no dependence on the control parameter c. It is therefore reasonable to include the term cij as given in
(2a). The term f(ij) is introduced to model how predator i divides its effort hunting the set of prey. This is necessary to see cosurvival of predator and prey species in the model as the number of species that initially occupy the ecosystem increases. Its
exact definition is considered later.
The two generalised equations now allow for m predators and n prey to occupy the ecosystem, with each prey having an
individual r value corresponding to its growth rate, and a cij term, which measures the predatorial effectiveness of the ith predator
at predating upon the jth prey.
III.
INITIAL SIMULATION AND RESULTS
The r and c control parameter values that are used in the model have been generated randomly, with maximum values being
defined as rmax and cmax. These values are randomly selected from uniform distributions over the intervals [0, rmax] and [0,cmax].
A focus has been placed on identifying species diversity (the number of surviving predator or prey species) in the model based
on the rmax or cmax value that is utilised.
The model has been executed for 100 different randomisations of the c and r control parameters for each value of rmax and cmax. It
is then executed for 5000 time steps for each set of control parameter settings. The average number of surviving predator and
prey species is recorded for each separate rmax and cmax value. Averaging over 100 different choices for the control parameter
values and executing for 5000 time steps has been found to be sufficient as there is no significant difference attained when
averaging over 1000 separate random choices or executing for 50,000 time steps. For the largest executed ecosystems averaging
over more than 100 separate parameter choices or running for more than 5000 time steps would be too computationally intensive
and therefore not viable. The execution of an n=m =100 ecosystem on an Intel Sandy Bridge i7 processor has a total execution
time of 1 hour when executed on a single thread.
An initial population size of 0.5 is used for all predators and prey in all the runs in this paper and if a population size falls below
ε =10-6 then its size is set to zero. The ε value that is used in the model can affect the results of the algorithm due to the use of the
Ricker model within the prey function. In the Ricker model the population sizes will never fall below zero, but for large values of
r part of the attractor will fall within the range [0,ε] . The r value at which the Ricker model will collapse therefore differs as ε
changes. With the use of other ε values similar behaviour is seen in the multiple species predator-prey model as with ε = 10-6.
A. Simple Predation Strategy
The simplest way to model a predator predating on a set of n prey is to assume that it spends an equal amount of effort predating
on each. In terms of the f(ij) term introduced in (2), this can be written as
(𝑖𝑗)
𝑓𝑎
1
= .
𝑛
(3)
3
Results detailing survival rates have been gathered for the model using this simple predation strategy for n=m =2, n=m =10,
n=m =100and n=m =1000 ecosystems.
B. Results and Analysis of Results
Figure 1 shows the survival rates for both the predators and the prey for a n=m =2, n=m =10 and n=m =100 ecosystem. There are
some areas of similarity in the rmax-cmax space across all of these graphs. For example, it can be seen that when the cmax < 1 there
is no predator survival in the model irrespective of the value of n and m.. Within this range the cij values are too low to support
predator survival. When the predators die out the prey survive independently of the predators, acting as uncoupled Ricker models
controlled by their individual r control parameter. For this reason, the same results are obtained for prey survival in this area
independent of the initial number of predator-prey species, since the predators are quickly dying off (within 10 generations),
removing any dependence on the cij control parameter values. Furthermore, as rmax increases above 4, the prey survival rate
decreases. The reason for this is that in the Ricker model the population of prey with r values greater than 4 will fall below the
cut-off value of ε =10-6. Despite this common behaviour for low cmax, the lowest value of cmax for which predator survival occurs
varies with n and m. For n=m =2, no predator survival occurs below cmax = 1, whereas for n=m =10, it is cmax = 1.3 and for n=m
=100 it is cmax = 1.7. An explanation for these results is discussed later. .
Figure 2 shows time series data within this region, with cmax = 1 and rmax = 1 for an n=m =10 ecosystem. It can be seen that the
predators all die off quickly, with their population sizes falling below ε, with the prey then beginning to act as uncoupled Ricker
models, all converging on a single attractor where their population size = 1.
The area of most interest is the area of co-survival in the (rmax,cmax) space. Figure 1a shows the results for a n=m =2 model. Here
the majority of predator survival can be observed between cmax = 1 and cmax = 9.5 and rmax = 0.3 and rmax = 9 For cmax greater than
1 there is some predator survival over the entire range of cmax and rmax values that have been executed.
Figure 1b shows the results for an n=m =10 ecosystem. The majority of predator survival is taking place between cmax = 1.3 and
cmax = 11.2 and rmax = 0.3 and rmax = 7.8 with a small amount of predator survival happening outside of this range. Note that the
region in which predator survival takes place is smaller in this case than for the n=m =2 case. It can now also be seen that the
survival of the predators has a detrimental effect on the survival of the prey. Within the area of co-survival predators are
managing to kill off some of the prey, reducing the survival rate of the prey. In the earlier 2x2 system this was not the case.
Figure 1c shows the results for an n=m =100 ecosystem. Predator survival is taking place between cmax = 1.7 and cmax = 11.2 and
rmax = 0.3 and rmax = 7.8 with a small. In fact, results for larger systems with numbers of predators and prey greater than 100
show that the rmax-cmax space in which predator survival takes place does not change further and so has converged for n=m =100.
The minimum cmax value at which co-survival takes place is also fixed at 1.7 for n=m=100 ecosystems. A very visible impact can
be seen here of predator survival causing a decline in the survival rates of the prey species since it is possible to see the shape of
the surviving predator survival space in that of the prey survival space.
Table 1 shows the final survival rates for both the predators and the prey. The effect that the number of species that initially
populate the ecosystem has upon the peak position of predator survival can be seen. For the smaller initial number of species (
n=m <200) the position of the peak changes as the number of species increases. Upon reaching n=m =200 the position of this
peak has become fully converged, now remaining fixed as n=m>200. It can also be noted that as the initial number of predators
and prey increases, the rate of predator survival is declining, yet the total number of predators that are surviving at the peak
positing is actually increasing.
Fig. 1c shows that for an n=m =100 ecosystem and values of rmax between 2 and 4, the predator survival rate increases as cmax
increases to area region in which the system can sustain the highest rate, and then very quickly drops off to zero. For example, at
rmax = 3.8, there is a large predator survival rate of 0.14 with cmax = 10.4, but it drops to almost zero when cmax = 11.2. Figure 4
shows the summation of all surviving predator populations for the first 50 time steps showing how this collapse is taking place.
Figure 4A shows this for the position cmax = 10.4, rmax = 3.8, where there is an initial growth in the total predator populations
followed by a sudden collapse (this is caused by the abundance of predators overhunting and depleting their prey), however the
population values are not all falling below ε. The small yet still existing predator population sizes allow the prey populations to
recover followed by the recovery of the still surviving predator populations. Figure 4.B shows that at the position cmax = 11.2,
rmax = 3.8 all predator populations very quickly fall below ε during the initial drop-off, causing total extinction of the predator
populations with the prey surviving independently.
4
C. Lowest value of cmax for which predator survival occurs
𝑐𝑖𝑗
(𝑗)
With the use of (3) as the predation strategy, the population of predator i decreases unless ∑𝑛𝑗=1 𝑁𝑡 ≥ 1. In order to
𝑛
determine the lowest value of 𝑐𝑚𝑎𝑥 at which predator survival occurs, it can be noted that in the absence of predators the prey are
represented by uncoupled Ricker models with a fixed point at 𝑁 (𝑗) = 1. This fixed point is stable for 𝑟𝑗 < 2, but then a
bifurcation point is reached and the system becomes periodic before becoming chaotic for 𝑟𝑗 > 2.7 with only narrow intervals of
periodic behaviour. However, for 𝑟𝑗 > 3.96 the population will fall below the cut-off value of 10-6 and hence be set to zero. Here
(𝑗)
the assumption will be made that that 𝑁𝑡 = 1 for all j such that 𝑟𝑗 < 3.96. This assumption is clearly valid for 𝑟𝑗 < 2, but it is
also valid for the range 2 ≤ 𝑟𝑗 < 3.96 since in this region it can be shown (see appendix ) that for the Ricker model the average
post-transient population is 1for an orbit of arbitrary period p . Based on this assumption, the condition for predator survival
𝑐𝑖𝑗
𝑛
becomes ∑𝑛𝑗=1
≥ 1 or, equivalently, ∑𝑛𝑗=1 𝑎𝑖𝑗 ≥
where 𝑎𝑖𝑗 are randomly selected from the uniform distribution on the
𝑛
𝑐𝑚𝑎𝑥
interval [0,1]. For n = 2, it is clear that survival cannot occur unless 𝑐𝑚𝑎𝑥 ≥ 1, which corresponds to figure 1a.
In general, for m predators and 100 runs the condition for the expected number of surviving predators to be less than 1 is:
Pr (∑𝑛𝑗=1 𝑎𝑖𝑗 ≥
𝑛
𝑐max
)≤
1
(4)
100𝑚
or equivalently,
Pr (∑𝑛𝑗=1 𝑎𝑖𝑗 ≤
𝑛
𝑐𝑚𝑎𝑥
)≥1−
1
(5)
100𝑚
The Irwin-Hall distribution gives the probability distribution for the sum of independent uniform random variables on the
interval [0,1]. Hence to determine the value for the LHS of (A.2), its cumulative distribution function can be used to obtain:
𝑛
Pr (∑𝑛𝑗=1 𝑎𝑖𝑗 ≤
𝑛
𝑐𝑚𝑎𝑥
)=
𝑛
⌊
⌋
𝑛
𝑛
∑ 𝑐𝑚𝑎𝑥 (−1)𝑘 ( ) (
− 𝑘)
𝑛! 𝑘=0
𝑐
𝑘
𝑚𝑎𝑥
1
(6)
Setting 𝑛 = 𝑚 = 10 and 𝑐𝑚𝑎𝑥 = 1.29 gives a value of 0.999 and so, via (A.2), survival would not be expected for values of
𝑐𝑚𝑎𝑥 < 1.29, which corresponds to the results in figure 1b (where results have been plotted in intervals of 0.1).
For large n the Irwin-Hall distribution can be approximated by a Gaussian distribution with 𝜇 = 𝑛/2 and 𝜎 2 = 𝑛/12. Using the
cumulative distribution for the Gaussian distribution (A.3) can be replaced by:
Pr (∑𝑛𝑗=1 𝑎𝑖𝑗 ≤
𝑛
𝑐𝑚𝑎𝑥
1
) ≈ [1 + 𝑒𝑟𝑓 (
2
𝑛
𝑛
−
𝑐𝑚𝑎𝑥 2
√𝑛⁄6
)]
(7)
where erf is the error function. For n=m=100, combining (A.2) and (A.4) gives the following condition for predator survival:
1
2
[1 + 𝑒𝑟𝑓 (
𝑛
𝑛
−
𝑐𝑚𝑎𝑥 2
√𝑛⁄6
)] ≥ 0.9999
(8)
which can be solved to obtain 𝑐𝑚𝑎𝑥 = 1.65 which corresponds with the results in figure 1c. For n=m=1000, a value of 𝑐𝑚𝑎𝑥 =
1.86 is obtained.
This can be used to explain a further feature of the results. As noted above, this approach breaks down for 𝑟𝑚𝑎𝑥 > 3.96 where the
lowest value of 𝑐𝑚𝑎𝑥 supporting predator survival increases with 𝑟𝑚𝑎𝑥 . This can be seen in each of the predator survival plots in
𝑐𝑖𝑗
figure 1. Since prey with r values greater than 3.96 will not survive, the condition for predators to survive becomes ∑𝑠𝑗=1
≥1
𝑛
where the sum is now over the number of surviving prey, s. Clearly, the corresponding values 𝑐𝑖𝑗 and hence 𝑐𝑚𝑎𝑥 need to be
greater for predators to survive.
D. Identification of which Predators and Prey are Surviving Based on their Control Parameters
5
For the larger initial number of species a structured uniform distribution of the r control parameter has been introduced in which
the r control parameters that are used in the model are distributed between 0 and rmax via
rj 
j
rmax .
n
(9)
For large n this approaches a uniform random distribution over [0,rmax].
Figure 5 shows the results from the model with the utilisation of this uniformly structured set of r control parameter values for
an n=m=100 ecosystem. These results confirm that the results achieved when structuring the r control parameters are the same as
the results where the r control parameters are assigned randomly.
As the structured assignment of the r control parameter value allows the easy identification of which r values belong to which
prey, it can be used to show exactly which prey are dying off using their individual r values.
An effectiveness measure has also been designed for the predators which defines the overall effectiveness of each of the
predators in the ecosystem:
𝑛
𝑝𝑟𝑒𝑑
𝐸𝑖
= ∑(𝑐𝑖𝑗 × 𝑟𝑗 )
(10)
𝑗=1
A summation is made of each of the predators cij values multiplied by the corresponding prey r value.
Figure 6a and 6b presents results for rmax = 3.8 and a cmax = 9.9, which corresponds to the peak position of predator survival for
an n=m=100 ecosystem. We make 1000 separate choices for the cij matrix, with the r values being uniformly structured between
0 and rmax. Figure 6a shows the survival rate averaged over these 1000 runs for each of the individual prey as identified by their r
values. It can be seen that with an r value below 0.5 the prey species always become extinct, with the possibility of survival
beginning as the r value grows greater than 0.5. The rate of survival increases until the r value becomes greater than r= 1.7,
where the prey survive in all runs.
𝑝𝑟𝑒𝑑
Figure 6b shows the survival rate of the predators as a function of their 𝐸𝑖
value as specified in (5). The effectiveness measure
has been calculated for each of the predators over the 1000 runs, with a record then being kept of whether the predator has
𝑝𝑟𝑒𝑑
survived or become extinct. Three plots are shown: the number of predator species that have become extinct for each 𝐸𝑖
𝑝𝑟𝑒𝑑
value, the number of species that have survived for each 𝐸𝑖
value and a summation of these, which shows the entire range of
𝑝𝑟𝑒𝑑
𝑝𝑟𝑒𝑑
𝐸𝑖
values. It can be seen that those predators with the lowest 𝐸𝑖
value, between 700 and 910, always die off. Once the
𝑝𝑟𝑒𝑑
𝑝𝑟𝑒𝑑
𝑝𝑟𝑒𝑑
𝐸𝑖
value becomes greater than 910 some predators now have the chance to survive. Between 𝐸𝑖
= 910 and 𝐸𝑖
= 1010,
𝑝𝑟𝑒𝑑
the majority of predators will become extinct, with some surviving. Once 𝐸𝑖
becomes greater than 1010 more predators
𝑝𝑟𝑒𝑑
𝑝𝑟𝑒𝑑
𝑝𝑟𝑒𝑑
survive than become extinct, with the chance of survival increasing as 𝐸𝑖
increases. For the highest values of 𝐸𝑖
(𝐸𝑖
>
1100), all predators survive.
Figure 6c and 6d shows corresponding results for rmax = 5 and cmax = 8.7, with figure 6c showing the survival rate of the prey and
𝑝𝑟𝑒𝑑
figure 6d showing the number of surviving predators as a function of their 𝐸𝑖
value. In figure 6c the effect of the cut-off
value ε on individual prey populations can be seen. The same sort of behaviour is initially present as that in figure 6a, with the
prey populations with the lowest r value dying off, and those within the range r = 2 to r = 4 always surviving and with a period
in-between in which the prey can either die off or survive. However, in figure 6c, once the r value grows greater than 4 there is a
very sudden fall off and collapse where the prey populations are guaranteed once again to die off. This is due to the use of the
Ricker model and the cut-off value as mentioned earlier. With an r value within this range the prey is guaranteed to eventually
fall below ε and is therefore guaranteed to die off. The predators here are behaving in a very similar way, with survival only
𝑝𝑟𝑒𝑑
being possible for those with the highest 𝐸𝑖
value. Similar behaviour is apparent across the entire (rmax,cmax) region of co𝑝𝑟𝑒𝑑
survival, with the prey with r values closest to the Ricker models cut-off point (r=3.9), and the predators with the highest 𝐸𝑖
values being the species most likely to survive. ε]
6
With the use of (3) as the predation strategy in all areas of co-survival both the prey with the lowest r value and the prey whose r
𝑝𝑟𝑒𝑑
𝑝𝑟𝑒𝑑
values are too high due ε become extinct. The predators with the lowest 𝐸𝑖
always die off, and those with the highest 𝐸𝑖
value always survive, with a subset in-between that will either die off or survive.
IV.
ALTERNATIVE PREDATION STRATEGIES
As noted earlier there is currently a simple predation strategy, where the cij matrix is being scaled by the initial number of prey
species that occupy the ecosystem. This corresponds to the predator dividing its time equally amongst each of the prey species
that initially populate the ecosystem. In this method of modelling the predators are effectively unaware that a prey species has
died off; in the event of a prey species dying off they continue to hunt for this extinct prey species.
For this reason several other predation strategies have been modelled. The first of these is where the predator species are aware
that a prey species has become extinct, and therefore redistribute their time over the remaining surviving prey. It is expressed as:
1
(𝑖𝑗)
𝑓𝑏
={
𝑗 ∈ 𝐿(𝑡)
|𝐿(𝑡)|
0
(11)
𝑗 ∉ 𝐿(𝑡)
where L(t) is the set of surviving prey at time step t. This is in effect a smarter prey, distributing its time over prey species which
continue to exist. Note also that in this model the predators are distributing their time dynamically over the set of surviving prey,
updating at each time step how many prey species still exist.
A further predation strategy where the predators focus their efforts upon the prey on which they are best at predating is modelled
via:
(𝑖𝑗)
𝑓𝑐
=
𝑐𝑖𝑗
𝑛
∑𝑗=1 𝑐𝑖𝑗
(12)
where the f(ij) values are scaled towards the highest cij value. In comparison to the original simple predation strategy. (3), this
will increase the amount of effort in predating on prey with the largest cij value but decrease the value for the smallest cij values
As with the original predation strategy (3), (12) does not vary with time. However, the combination of (11) and (12), in which
the predators both focuses their efforts and are aware that a prey has become extinct can be formulated as:
𝑐𝑖𝑗
(𝑖𝑗)
𝑓𝑑
={
∑𝑛𝑗∈𝐿(𝑡) 𝑐𝑖𝑗
0
𝑗 ∈ 𝐿(𝑡)
(13)
𝑗 ∉ 𝐿(𝑡)
Here the cij values are dynamically scaled over the set of surviving prey and so this is in effect a smart and focused predator.
Two further predation strategies may also be considered. Firstly there is the case where the predators focus their efforts predating
upon those prey with higher r values:
𝑟𝑗
(𝑖𝑗)
(14)
𝑓𝑒 =
𝑛
∑𝑗=1 𝑟𝑗
A dynamic version of this function can also be modelled in which they focus their effort towards prey with the highest r value
amongst the set of surviving prey:
𝑟𝑗
(𝑖𝑗)
𝑓𝑓
= { ∑𝑗∈𝐿(𝑡) 𝑟𝑗
0
𝑗 ∈ 𝐿(𝑡)
𝑗 ∉ 𝐿(𝑡)
(15)
7
A final predation model is where the predators focus their efforts upon the prey with the highest current population size:
(𝑗)
(𝑖𝑗)
𝑓𝑔
𝑁𝑡
={
(𝑗)
∑𝑗∈𝐿(𝑡) 𝑁𝑡
0
𝑗 ∈ 𝐿(𝑡)
(16)
𝑗 ∉ 𝐿(𝑡)
It can be seen here that the f(ij) matrix is composed of the current prey population of an individual prey divided by the summation
of all the prey population. When this is then multiplied by the cij matrix this will scale each of the cij values towards the
corresponding prey current population size.
V.
COMPARISON OF PREDATION STRATEGIES
Figures 8-13 show the survival rates across the rmax-cmax space for an n=m=100 ecosystem for each of the predation strategies
outlined in section IV. These figures can each be directly compared to figure 1c, which shows results for an n=m=100 system
using the simple predation strategy.
Figure 8 shows the results obtained using (11), which models the smarter predators which are aware that a predator has died off.
This is the most similar of the predation strategies to that of the original model, with the difference being that a predator
dynamically recalculates the f(ij) matrix in the event of a prey dying off. This contrasts with the simple predation strategy (3),
where the f(ij) matrix is calculated at the start, and the values then remaining fixed as the model runs. The similarity in the models
is reflected in the overall area of co-survival being very similar between (3) and (11). The only significant difference in the space
in which both predators and prey co-exist occurs for higher values of cmax. For values of rmax lower than 4 and all values of cmax,
the region of co-survival is very similar. However, as can be seen in figure 1c, as rmax grows greater than 4, with (3), there is an
area that begins to develop where all predators begin to die off within the region cmax = 1.7-4. Here some prey become extinct not
due to hunting, but due to their r value being greater than 4, therefore their populations are guaranteed to fall below ε. The prey
that do survive cannot sustain the predator species survival. In figure 8, where predators are now aware that a predator has died
off, this region of predator extinction has disappeared. In this model at least some predator survival occurs for cmax = 1.7 and for
all values of rmax between 0.3 and 10. The ability for the predators to focus on only the prey that survive allows survival in this
region.
A more detailed explanation for this behaviour can be found by considering again the discussion in III.C since the lowest value
of 𝑐𝑚𝑎𝑥 at which predators can survive remains the same for 𝑟𝑚𝑎𝑥 > 4. The reason for this is that the predators adjust their
behaviour to take account of prey becoming extinct in a way which is equivalent to increasing 𝑐𝑖𝑗 , and hence for greater values of
𝑟𝑚𝑎𝑥 a lower value of 𝑐𝑚𝑎𝑥 is able to support predator survival than was the case using the initial predation strategy. In fact, the
𝑐𝑖𝑗
condition for predators to survive becomes ∑𝑠𝑗=1
≥ 1 where the sum is now over the number of surviving prey, s, which is
𝑠
very similar to the expression used in III.C.
Table 2 shows the position of peak predator species survival of the model defined by (11) for different initial numbers of species
as specified by n and m. For a n=m=100 ecosystem there is a predator survival rate of 0.164 at the point of peak predator
survival, where cmax = 7.5 and rmax = 2.5.At the same point with the use of the simple predation strategy there is a predator
survival rate of 0.13. The peak survival rate with the use of (11) is also slightly higher than the peak survival rate of 0.153
𝑝𝑟𝑒𝑑
obtained using (3) (see Table 1). Figure 14a shows the number of surviving predators based on their 𝐸𝑖
value at the peak cmaxrmax position of predator survival, with the broken line showing the number of predators that have become extinct, the solid line
𝑝𝑟𝑒𝑑
showing the predators that have survived and the dotted line showing the entire range of predator 𝐸𝑖
values. Similar to the
𝑝𝑟𝑒𝑑
simple predation strategy it is the predators with the highest 𝐸𝑖
values that are most likely to survive in the ecosystem.
As has been noted, the area in which predator and prey species can co-exist between the two predation strategies is similar,
however the survival rates within this area are significantly different. The peak position of predator survival itself has shifted to
cmax = 7.5 and rmax = 2.5, compared to cmax = 9.9 and rmax = 3.8 when using (3). For values of cmax between cmax = 8 and cmax =12
an area has developed in which there is very little predator survival. Figure 14b groups similar predators based on their overall
𝑝𝑟𝑒𝑑
𝐸𝑖
value defined in (10), and shows which of the predators survive for a point within the region of supressed predator
survival (cmax = 9.9, rmax = 4.6). The predator survival rate at this point is 0.009 and the prey survival rate is 0.38 when (11) is
used for the predation strategy. By contrast, at this point with the simple predation strategy (3) there is a predator survival rate of
𝑝𝑟𝑒𝑑
0.12 and a prey survival rate of 0.6. On average a single dominant predator survives at this point. The total range of 𝐸𝑖
values
𝑝𝑟𝑒𝑑
𝑝𝑟𝑒𝑑
𝑝𝑟𝑒𝑑
at this point is between 𝐸𝑖
= 800 and 𝐸𝑖
= 1400. In figure 14.b it can be seen that predator survival occurs between 𝐸𝑖
𝑝𝑟𝑒𝑑
𝑝𝑟𝑒𝑑
= 1070 and 𝐸𝑖
= 1400, with a very low rate of survival within this range, unlike figure 14(a), there is no region of 𝐸𝑖
values where the predators are guaranteed to survive. Figure 15 shows the underlying behaviour of the predator and prey species
8
at this point. Figure 15a shows the survival rate of the prey and predator species for each time step and 15b shows the summation
of each of the predator and prey populations at each time step. It can be seen in 15b there is significant variation in the transient
behaviour of the predator and prey total population sizes over the first 100 time steps, where the predator species are overhunting
the prey species. This is reflected in figure 15a, which shows that the predators die off within the first 100 time steps, leaving on
average a single dominant predator in the ecosystem.
Figure 9 shows the results obtained when (12) is used as a predation strategy. This equation models a focused predator, which
focuses its efforts predating the prey that benefit it the most (those with the highest corresponding cij value). Like the simple
predation strategy, this only scales the cij values before the execution of the model, with the values then remaining fixed as the
model runs. The rmax-cmax space in which co-survival of predators and prey is possible has decreased to lower values of cmax in
comparison to the use of the simple predation strategy, The behaviour as rmax increases is similar. Co-existence between the
predators and prey occurs between cmax = 1.4 – 8.5 and rmax = 0.3 – 9, with a small amount of predator species surviving at the
boundaries of this region, whereas in the case of the simple predation strategy, predator survival extends out to cmax = 11.2. For a
cmax value of lower than 1.4 the same behaviour is apparent in both models. In this region, the predators are not effective enough
to have any impact on the prey before becoming extinct, leaving uncoupled Ricker models that behave independently of the
predators. In this model however there is a chance of predator survival between cmax = 1.4 and cmax = 1.7. In this region in (3) and
(11) predator extinction is guaranteed.
The overall shape of the area in which co-survival can take place is visually similar to the simple predation strategy, but
compressed in the cmax direction. Although the area in which survival can take place has declined in size, the rate of predator
survival within this smaller region has increased. Table 3 shows the survival rates of the predator and prey species at the position
of peak predator survival. It can be seen that for an n=m=100 system this peak position is at cmax = 7.4, rmax = 3.3, with a predator
survival rate of 0.273, and a prey survival rate of 0.75. The predator survival rate here is much higher than the simple predation
strategy, which has a peak predator survival rate of 0.153. This shows that allowing the predators to focus on the prey that
benefit them the most can generate more predator survival than where they evenly distribute their time over each of the prey
species.
Figure 10 shows the results obtained when (13) is used as a predation strategy. This equation models a predator that is both
focused on the prey that benefit it the most, and is aware that a prey species has become extinct, redistributing its efforts across
the remaining prey species. Similar to how the predation strategy outlined in (11) is an adaptive form of (3), (13) is an adaptive
form of (12), with the same behavioural changes developing between (13) and (12) as those that have been outlined for (11). The
overall rmax-cmax space in which co-survival takes place is very similar between (13) and (12), with the only difference again
being the disappearance of a region of predator extinction for low cmax and high rmax values that was discussed in the case of (11).
The cmax for which co-survival is first seen is cmax = 1.4, the same as when (12) was used.
Once again, however, although the space is very similar, the ability of the predators to redistribute their effort in the event of a
prey species becoming extinct has introduced a region of suppressed predator survival between cmax = 6 and cmax = 8.5. The same
survival behaviour is occurring in this region as has been outlined in in the discussion of (11).
The peak position of predator survival has shifted to a lower value of cmax and the survival rate has increased slightly in
comparison to (12), in a similar way to how the peak position and survival rate in (11) shifted in comparison to (3). As shown in
in table 4 the peak position of predator survival is occurring at cmax = 5.9 and rmax = 3.2, with a predator survival rate of 0.304 and
prey survival rate of 0.75. Out of all the predation strategies that have been used in this paper, this is the one that produces the
maximum survival rate for the predators.
The change in behaviour found when the predation strategy specified in (13) is compared with that in (12) is very similar to the
change between the strategy specified in (11) and (3). This shows that these differences are brought about as a result of the
predators being able to refocus their efforts in the event of a prey becoming extinct.
Concerning the lowest cmax value at which predators can survive that is discussed in detail for (3) in subsection III.C, the same
effect is seen in figure 10 with the use of (13) as the predation strategy as outlined for when (11) is used. Both of these models
employ predation strategies that adjust for prey becoming extinct and so the lowest value of cmax at which predators can survive
remains the same for higher values of rmax as it is for low values of rmax.
Figure 11 shows the results obtained when (14) is used as a predation strategy. This equation models predators that focus their
efforts predating upon those prey with the highest r value, the prey which have the greatest growth rate within the model. In this
case since the prey r values are distributed evenly between 0 and rmax, each predator will scale its predatorial effort towards each
of the prey in the same way. Here co-survival of the predators and prey takes place between cmax = 1.8 – 12.5 and rmax = 0.3-9.6.
In comparison to the earlier predation models, the space in which co-survival can take place fills a diagonal area through the rmaxcmax space. Between the point cmax = 1.8, rmax = 4 and cmax = 9,rmax = 9.3 a line can be drawn above which predator extinction
9
quickly takes place. Above this line the predators are initially overhunting the prey to the point that the ecosystem cannot support
any predators, however prey continue to survive as their populations do not fall below ε allowing them to recover after predator
extinction. In all the earlier predation strategies co-survival can takes place within this area.
Unlike the previous predation strategies, for (14), a bounded area has developed within the co-survival rmax-cmax space where
almost all predators become extinct. This region is visible in Figure 11 between cmax = 5.9, rmax = 7.3 and cmax =8.7,,rmax =7.7.
Figure 16 shows the behaviour of the predator and prey for a point in this region with cmax = 7.7 and rmax = 7.4. At this point there
is a predator survival rate of 0.004 and a prey survival rate of 0.38. Figure 16a shows the average survival rate of the predators
and prey over the first 250 time steps of the model, the prey being represented by the broken line and the predators the solid line.
It can be seen that within the first 10 time steps, the majority of predators quickly die off, leaving a predator survival rate of 0.1.
Over the next 240 time steps they slowly decline to a survival rate of 0.004, where the survival rate levels out. The prey survival
rate converges on a value of 0.38 within the first five time steps and then remains fixed as the model runs. Within this area if any
prey dies off it will be within the first 10 time steps of execution. Figure 16b shows the summation of both the predator and prey
populations over the first 50 time steps. Erratic behaviour is occurring over the first 10 time steps and as noted this is the area
where the majority of the predators and the prey die off. After this the predators’ total population size settles on 1.9 and the total
prey population size settles on 31, with very little change between the various time steps. This suggests that the majority of
surviving species have converged upon a single population size within the model.
The behaviour of the prey in this model is also very different to that of the other predation strategies. Outside of the region of cosurvival, or the bounded region, there is nowhere within the tested rmax-cmax space that complete prey extinction is taking place.
With high cmax values (cmax > 12) the predators are depleting some of the prey, but not all of them. The ability for the predators to
focus on the prey with the highest r value is allowing some prey in this region to continue to survive, where in other models they
would be guaranteed to die off. Figure 17 shows the behaviour of the prey within this region. Figure 17a shows the summation of
all the population sizes for both the predator and the prey over the first 50 time steps. The prey is the broken line and the predator
is the solid line.
It can be seen that over the first five iterations the predator populations hunt the prey populations down to the point where all
species are almost extinct. This depletion in the predators’ food source results in the predators becoming extinct themselves, with
the surviving prey then recovering as the model continues to run. Figure 17b shows which prey have survived based on their r
value. It can be seen here that unlike in the previous predation strategies, it is the prey with the lower range of r values that are
surviving. In all previous models it is prey with the highest r values that die off which could explain the differences in species
survival between (14) and the earlier strategies. In (14) the predators are focusing their effort 1predating upon prey with the
highest r value, while the prey with lower r values receive comparatively less predation and therefore are more likely to survive.
Table 5 shows the final survival rates for the rmax-cmax position of peak predator survival when (14) is used for the predation
strategy. It can be seen that in an n=m= 100 ecosystem the peak position of predator survival is at cmax = 8.5 and rmax = 1.9. At
this point there is a predator survival rate of 0.19 and a prey survival rate of 0.94. This predator survival rate is higher than it is
for (3) and (11), but lower than the models that focus towards the cij values ((12) and (13)). The prey survival rate at the point of
peak predator survival is higher in this model than all the previous models, with a prey survival rate of 0.94. The predators are
only managing to kill off an average of 6 prey in each run. Similar to in figure 17b, and unlike any of the other models, it is
always the prey with the highest r values that are being hunted to extinction in this model.
Similar to the behaviour discussed for (3) in subsection III.C, in figure 11, where the strategy in (14) is used, the lowest values of
𝑐𝑚𝑎𝑥 supporting predator survival are the same as the standard case for low values of 𝑟𝑚𝑎𝑥 (𝑟𝑚𝑎𝑥 < 4), but for higher values of
𝑟𝑚𝑎𝑥 the lowest values of 𝑐𝑚𝑎𝑥 required for survival are greater than in the standard case. The explanation for this is that this
strategy effectively scales 𝑐𝑖𝑗 by 𝑟𝑗 which for higher values of 𝑟𝑚𝑎𝑥 means that the values of 𝑐𝑖𝑗 corresponding to surviving prey
(i.e. those with low r values) are effectively reduced and so a greater value of 𝑐𝑚𝑎𝑥 is needed for predator survival.
Figure 12 shows results obtained when (15) is used as a predation strategy. This equation models a predator that is both focused
on the prey that have the highest r values, and is aware that a prey species has become extinct, redistributing its efforts across the
remaining prey species. Similar to how the predation strategy outlined in (11) is an adaptive form of (3), (15) is an adaptive form
of (14). Unlike in the previous models, where the overall space in which co-survival takes place was very similar between the
adaptive and non-adaptive form of predation strategy, the space has significantly changed between figure 11 and figure 12. The
area discussed above the line between cmax = 1.8 cmax = 4 and cmax = 9,rmax =9.3, where the predators are all dying off, has
disappeared. Similar to the other models that can adapt when a prey dies off, predator survival occurs for values of cmax ≥ 1.7 for
all values of rmax between 4 and 10. This suggests the ability to adapt is what allows co-survival to take for the lowest cmax values
at which survival takes place and the higher range of rmax values (rmax>4).
10
Similar to the results obtained using the predation strategy in (14), a bounded area with very little predator and prey survival has
developed within the region of co-survival rmax-cmax space. This area occurs around the same area as in figure 11a, but has
become more clearly defined. The behaviour in this area is the same as that discussed for the predation strategy in (14).
Once again, similar to the results obtained using (11) and (13) which also dynamically update the f(ij) matrix in the event of a prey
dying off, an area within the rmax-cmax space of co-survival has formed with suppressed predator survival, which occurs for
cmax>7.3. This confirms that the ability to adapt when a prey becomes extinct introduces this region within the area of cosurvival.
Table 6 shows the survival rates for the rmax-cmax position of peak predator survival when (15) is used for the predation strategy.
It can be seen that in an n=m= 100 ecosystem the peak position of predator survival is at cmax = 6.7 and rmax = 2.8. At this point
there is a predator survival rate of 0.18 and a prey survival rate of 0.94. This is slightly lower than the peak rate of survival when
(14) was used. This differs from the other strategies which adapt when prey become extinct, i.e. strategies (13) and (11), which
had a larger peak survival rate compared to the strategy using (12) and (3). This shows that the ability for the predators to adapt
when a prey dies off does not guarantee that there will be more peak predator survival.
Concerning the lowest cmax value at which predators can survive as discussed in detail for (3) in subsection III.C, the same effect
is seen in figure 12 with the use of (15) as in the case when the predation strategy in (11) is used. This is the case for (11), (13)
and (15). All three of these strategies adjust for prey becoming extinct.
Figure 13 shows results obtained when (16) is used as a predation strategy. (16) is a model where the predators focus on the prey
with the highest current population sizes. Since the population sizes of the species occupying the system changes at every
generation, this means that the f(ij) values have to be updated at every generation. This differs from all the previous models, in
that f(ij) is a function of the dynamically varying prey values, rather than the control parameters rj and cij . This accounts for
how different the rmax-cmax area of co-survival is in comparison to the earlier models. In this model co-survival takes place
between rmax = 0.3 – 10 and cmax = 1.1-9.5.
For rmax values between 0.3 and 2.6, co-survival only occurs in the model for cmax> 1.7. This is similar to the other models.
However, for rmax > 2.6, the lowest cmax value for which co-survival occurs, changes. Between rmax = 2.6 and rmax = 4, the lowest
cmax value for which co-survival can occur decreases slightly as rmax increases becoming static at rmax = 4. For rmax = 4 to rmax = 10
co-survival takes place for a cmax value of 1.1.
In Figure 13 it can be seen that there is a boundary of predator extinction along a line which can be drawn between cmax =8.7, rmax
= 2.2, and cmax = 2.3, rmax = 5.1. Unlike the other predation strategies, which have comparatively smooth boundaries between
extinction and co-survival, in figure 13a this boundary is fractured and jagged. The impact that the predator has on the prey in
this model is also apparent. This fractured region can also be seen in the prey survival, with a boundary in prey extinction
occurring along the same line. This is due to the predators overhunting the prey species.
Figure 18 above shows the survival rates of each of the prey based on their r value for two separate points in the rmax-cmax space
when (16) is used as the predation strategy. Figure 18a shows this for rmax = 4, cmax = 8, which is above the fractured boundary of
co-survival. At this point there is a prey survival rate of 0.15 and a predator survival rate of 0. The predators are overhunting the
prey to a point where they cannot sustain themselves, and are therefore dying off, leaving on average 15 surviving prey species.
In figure18a it can be seen that when prey are categorized based on their r value, no prey species is guaranteed to survive, with
very few prey species being guaranteed to become extinct. The only prey that always become extinct are those with r values
between 0.88 and 1.08. The prey most likely to survive have r values between 0.04 and 0.44, these have over a 50% chance of
survival. Above r = 1.08 prey survival is very erratic, with no clear structure to indicate which prey can survive. This is
significantly different from the earlier models which have a smooth line showing which prey survive or become extinct. Figure
18b shows this for rmax = 2, cmax = 8, which is below the fractured boundary of co-survival. At this point there is a prey survival
rate of 0.95 and a predator survival rate of 0.07. Here the prey with the highest r value (greater than 1.82) are those most likely to
die off. With an r value between 0.3 and 0.58 it is also possible for a prey species to become extinct, but all other prey species
are guaranteed to survive. This shows that even in the area of co-survival, prey survival based on the r value is not smooth with
the use of (16).
Table 7 shows the survival rates for the rmax-cmax position of peak predator survival with the use of (16). It can be seen that in an
n=m= 100 ecosystem the peak position of predator survival is at cmax = 4.7 and rmax = 1.2. At this point there is a predator
survival rate of 0.11 and a prey survival rate of 1. This is the lowest predator survival rate at the peak position of predator
survival of any of the predation strategies. Also, unlike any of the other predation strategies, the predators are failing to kill off
any of the prey species at this peak position.
11
Concerning the lowest cmax value at which predators can survive as discussed in detail for (3) in subsection III.C , the results in
figure 13 are more complicated since although the strategy in (16) adjusts for prey extinction as some of the other strategies do, it
takes into account the population size of the prey as well.
Table 8 shows survival data for the position of peak predator species survival and the position for which the average summation
of the predator populations is at its highest for an n=m=100 ecosystem with the use of each of the predation strategies. For
predation strategy (3) the point of peak predator diversity is occurring at cmax = 9.9, rmax =3.8, with a survival rate of 0.153,
however when all the surviving predator population are summed an average total predator population size of 20.1 is seen over
the 100 runs. This does not correspond to the point at which the system contains the highest total predator population, with this
occurring at cmax =4.9, rmax =4.5, where there is a higher total predator population size of 46.2, but with a significantly lower
predator survival rate of 0.09. Although less predator species are surviving at this point their population sizes within the model
are much higher. This is true for each of the predation algorithms investigated, with the point of peak diversity always a
considerable distance from the point of most survival. In fact, in all cases the peak of total predator population size is more than
twice as high its value at the point of peak diversity.
VI.
CONCLUSION
This paper has looked at a multispecies discrete time predator-prey model with different ways of modelling predation strategies
for the predators. A focus has been placed on ecosystems with a range of predator and prey species (as defined by their control
parameters) and the final predator and prey survival rates of the ecosystem have been examined after sufficient generations of
execution (5000) to ensure post transient behaviour has been reached.
Though a range of predation strategies have been considered, several similarities between them may be noted. , Thus for the
strategies defined by (3), (12) and (14) the point of peak predator diversity in the co-survival space is very close to the edge of
total collapse, with cmax only having to increase slightly for no predators to survive. Strategies (11),(13) and (15) all contain
smarter predators, which redistribute their effort in the event of a prey becoming extinct, all three of these predation strategies
exhibit similar changes to the region of co-survival when compared the predation strategies that do not dynamically redistribute
the predators efforts, with a large region of supressed predator survival and a fixed value of c max for which predator survival first
occurs arising.
Furthermore, an explanation has been provided for the lowest cmax value at which predators can survive for various numbers of
species and across some of the different predation strategies. Finally, it has also been shown that maximising species diversity in
the ecosystem does not correspond to maximising total survival since the point of peak predator species diversity in the region of
co-survival is always found to be a considerable distance away from the point of highest total population survival.
Possible directions for future work include the investigation of alternative models to the Ricker-based model of Neubert et al
employed here as well as alternative predation strategies and perhaps strategies that could be adopted by prey. A particular focus,
however, will be on the introduction of mutation between the various predators and prey in order to explore what effect this has
on the ecosystem.
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13
Appendix: Proof that the average value of the points on an attractor of period p for the Ricker model
is unity.
Consider the Ricker model defined by
xi 1  f ( xi )  xi exp(r (1  xi )) .
(A1)
For an attractor of period p we can write
x p1  f ( p ) ( x1 )  x1
(A2)
where f ( p ) ( x) is the pth-iterate of f.
Using (A1) we can write,
x1  f ( p ) ( x1 )  f ( f ( p 1) ( x1 ))  f ( p 1) ( x1 ) exp(r (1  f ( p 1) ( x1 ))
(A3)
f ( p 1) ( x1 )  f ( f ( p  2) ( x1 ))  f ( p  2) ( x1 ) exp(r (1  f ( p  2) ( x1 ))
(A4)
and then noting that
and substituting (A4) into (A3) and continuing recursively ultimately gives
p 1
x1  x1  exp(r (1  f (i ) ( xi )))
(A5)
1 p 1 (i )
 f ( x1 )  1 .
p i0
(A6)
i0
Which reduces to
Noting that
f
(i )
( x1 ) : i  0.. p  1 is the set of points on the attractor of period p the result is proved.
14
Fig 1a. Survival rates for a
n=m =2
Fig 1b. Survival rates for a
n=m =10
Fig 1c. Survival rates for a
n=m =100
Fig 1. Survival rates averaged over 100 runs of the model and 5000 time steps at each run for n =2 m = 2, n = 10 m = 10 and n = 100 m = 100. A greyscale
colour bar is provided to identify the rate of survival at each point. Note the difference in the ranges of the greyscales in each case.
15
Fig. 2. Time series plot for rmax = 1 and cmax = 1
In this region of rmax - cmax space all the predators are quickly dying off within 10 generations. This leaves 10 uncoupled Ricker models acting as the prey population. These have survived in the
absence of the predator populations
No.
Predator
No.
Prey
cmax
2
10
100
200
500
1000
2
10
100
200
500
1000
3.9
7
9.9
10.2
10.4
10.4
rmax Position
Predator Survival Rate
Prey Survival Rate
2.2
3
3.8
3.9
3.9
3.9
0.542
0.302
0.153
0.125
0.095
0.076
0.936
0.745
0.720
0.720
0.718
0.711
Position
Table 1. A table showing the behaviour of the model at the position of peak predator survival for various initial numbers of species
This table shows the survival rate of predators and prey at the area of peak predator survival in the rmax-cmax space for various initial numbers of species. It can be seen that as the number of species that initially
populate the ecosystem increases the position of the peak is becoming converged upon the point cmax = 10.4 and rmax = 3.9. This remains the case for higher initial numbers of species, showing that the rmax-cmax space
in which co-existence takes place has converged.
4a cmax = 10.4
4b cmax = 11.2
Fig. 4. A sharp drop off to total predator collapse as Cmax increases
This figure shows the total population size of all surviving predators for the first 50 time steps of the model for cmax = 10.4 and cmax = 11.2 for a n=m =100 ecosystem. In both cases rmax = 3.8. This shows how it
quickly drops off from a point of vast predator population survival to complete predator extinction.
16
Fig. 5. A Structured Assignment of the r Control Parameter
Figure 5 shows the survival rates for the n=m =100 system with the introduction of a structured r control parameters, with the r control parameter values being uniformly distributed between 0 and rmax for the 100
prey that occupy the ecosystem. This confirms that the same results are achieved with uniformly structured r values as in the case with the randomly generated r values.
6a rmax = 3.8 a cmax = 9.9
6c rmax = 4.8 a cmax = 8.7
6b rmax = 3.8 a cmax = 9.9
6d rmax = 4.8 a cmax = 8.7
Figure 6. Predator and prey species survival at two points in the rmax - cmax space
Figure 6a and 6b presents results for rmax = 3.8 and a cmax = 9.9, with 6a showing the survival rate for each of the individual prey as identified by their r values and 6b shows the survival rate of the
predators as a function of their 𝐸𝑖𝑝𝑟𝑒𝑑 value. 6c and 6d show the same rmax = 4.8 and a cmax = 8.7. In 6b and 6d Solid line = surviving predators, Broken line = extinct predators, Dotted line = Entire
range of 𝐸𝑖𝑝𝑟𝑒𝑑 values.
17
Fig 8. Survival rates for a n=m=100 ecosystem with the use of the predation strategy outlined in (11)
This figure shows survival rates after 100 runs of the model and 5000 time steps at each run for n = 100 m = 100 using the predation strategy outlined in (6). A
greyscale color bar is provided to identify the rate of survival at each point.
Fig 9. Survival rates for a a n=m=100 ecosystem with the use of the predation strategy outlined in (12)
This figure shows survival rates after 100 runs of the model and 5000 time steps at each run for n = 100 m = 100 using the predation strategy outlined in (7). A
greyscale color bar is provided to identify the rate of survival at each point..
Fig 10. Survival rates for a n=m=100 ecosystem with the use of the predation strategy outlined in (13)
This figure shows survival rates after 100 runs of the model and 5000 time steps at each run for n = 100 m = 100 using the predation strategy outlined in (13). A
greyscale color bar is provided to identify the rate of survival at each point.
18
Fig 11. Survival rates for a for a n=m=100 ecosystem with the use of the predation strategy outlined in (14)
This figure shows survival rates after 100 runs of the model and 5000 time steps at each run for n = 100 m = 100 using the predation strategy outlined in (14). A
greyscale color bar is provided to identify the rate of survival at each point.
Fig 12. Survival rates for a n=m=100 ecosystem with the use of the predation strategy outlined in (15)
This figure shows survival rates after 100 runs of the model and 5000 time steps at each run for n = 100 m = 100 using the predation strategy outlined in (15). A
greyscale color bar is provided to identify the rate of survival at each point.
Fig 13. Survival rates for a a n=m=100 ecosystem with the use of the predation strategy outlined in (16)
This figure shows survival rates after 100 runs of the model and 5000 time steps at each run for n = 100 m = 100 using the predation strategy outlined in (16). A
greyscale color bar is provided to identify the rate of survival at each point.
19
No.
Pred-ator
No.
Prey
cmax
2
10
100
200
500
2
10
100
200
500
3.8
5.9
7.5
7.7
7.6
rmax Position
Predator Survival Rate
Prey Survival Rate
2.6
2.9
2.5
3.8
3.9
0.537
0.33
0.164
0.13
0.094
0.938
0.73
0.733
0.17
0.7
Position
Table 2. A table showing the behaviour of the model at the position of peak predator survival for various initial numbers of species and using (11) for
the predation strategy.
14a
14b
Figure 14. Predator species survival at two points in the rmax - cmax space with the use of (11) for the predation strategy
This figure shows the number of predators that either become extinct at two separate points in the model based on their 𝐸𝑖𝑝𝑟𝑒𝑑 value calculated using (11) this sentence does not make sense to me..
14a is at the peak of predator survival, where cmax = 7.5 and rmax = 2.5. 14b is within the new region that has developed with the use of (11) as the predation strategy, where a single dominant
predator is surviving. Here cmax = 9.6 and rmax = 4.6. Solid line = surviving predators, Broken line = extinct predators, Dotted line = Entire range of 𝐸𝑖𝑝𝑟𝑒𝑑 values.
15a
15b
Figure 15. Predator and Prey species behaviour at cmax = 9.6 and rmax = 4.6
This figure shows the behaviour of the predators and prey with the use of (11) within the new area with a very low predator survival rate. 12a shows the survival rate of the prey(broken) and
predator(solid) as the model at each time step. 12b shows the summation of each of the surviving populations for both the predator and the prey populations at each time step.
20
No.
Pred-ator
No.
Prey
cmax
2
10
100
200
500
2
10
100
200
500
3.3
5.5
7.4
7.7
7.8
rmax Position
Predator Survival Rate
Prey Survival Rate
3.2
2.7
3.3
3.9
3.9
0.6
0.47
0.273
0.22
0.169
0.97
0.8
0.75
0.75
0.74
Position
Table 3. A table showing the behaviour of the model at the position of peak predator survival for various initial numbers of species and using (12) for
the predation strategy.
No.
Pred-ator
No.
Prey
cmax
2
10
100
200
500
2
10
100
200
500
3.6
4.9
5.9
5.8
5.8
rmax Position
Predator Survival Rate
Prey Survival Rate
2.6
3.2
3.2
3.8
3.9
0.6
0.48
0.304
0.25
0.20
0.96
0.8
0.75
0.74
0.70
Position
Table 4. A table showing the behaviour of the model at the position of peak predator survival for various initial numbers of species and using (13) for
the predation strategy.
16a
16b
Figure 16. Predator and Prey population behavior at cmax = 7.7 and rmax = 7.4 with (14)
This figure shows the behaviour of the predator and prey populations with the use of (14). 16a shows the survival rate of the prey(broken) and predator(solid) populations as the model at each time
step. 16b shows the summation of each of the surviving populations for both the predator and the prey populations at each time step.(Solid Line Predator, Broken Line Prey)
17a
17b
Figure 17. Prey populations behavior at cmax = 12.6 rmax = 1
This figure shows the prey populations behavior at maxC = 12.6 and maxR = 1. This is a region in which the prey populations survive with the use of (14), but in all the other tested predator
strategies become extinct. 17.a shows the behavior of the populations over the first 50 timeteps and 17b shows which prey are surviving based on their r value. (Solid Line Predator, Broken
Line Prey)
21
No.
Predator
No.
Prey
cmax
2
10
100
200
500
2
10
100
200
500
4
6
8
8.7
8.7
rmax Position
Predator Survival Rate
Prey Survival Rate
3.1
2.4
3.5
3.6
3.6
0.53
0.34
0.19
0.16
0.12
1
0.98
0.94
0.93
0.95
Position
Table 5. A table showing the behaviour of the model at the position of peak predator survival for various initial numbers of species and using (14) for
the predation strategy.
No.
Predator
No.
Prey
cmax
rmax Position
Predator Survival Rate
Prey Survival Rate
Position
2
2
4.2
3.2
0.53
1
2
2
6.2
2.8
0.33
0.96
100
100
6.7
2.8
0.18
0.94
200
200
6.9
3.9
0.145
0.99
500
500
6.9
3.9
0.11
0.98
Table 6. A table showing the behaviour of the model at the position of peak predator survival for various initial numbers of species and using (15) for
the predation strategy.
18a
Figure 18. Prey species survival with (16)
18b
This figure shows the survival rate of each prey based on its r value with the use of eq(16). 18a shows this for rmax = 4, cmax = 8 and 18b shows this for rmax = 2, cmax = 8.
22
No.
Pred-ator
No.
Prey
cmax
2
10
100
200
500
2
10
100
200
500
3.5
5
4.7
4.7
4.9
rmax Position
Predator Survival Rate
Prey Survival Rate
2.2
2
1.2
1.8
2.5
0.54
0.27
0.11
0.087
0.0647
1
1
1
1
1
Position
Table 7. A table showing the behaviour of the model at the position of peak predator survival for various initial numbers of species and using (16) for
the predation strategy.
cmax
rmax
Survival Rate
Total Survival
cmax
rmax Position –
Survival Rate
Total Survival
Position –
Position –
Position –Total
Peak Diversity
Peak
Peak
Survival Peak
Diversity
Diversity
(3)
9.9
3.8
0.153
20.1
4.9
4.5
0.09
46.2
(11)
7.5
2.5
0.164
10
3.8
4.6
0.09
46.2
(12)
7.4
3.3
0.273
20.6
3.6
4.7
0.158
48.2
(13)
5.9
3.2
0.304
13.4
3
4.6
0.18
48.7
(14)
8
3.5
0.19
19
3.4
5
0.08
58
(15)
6.7
2.8
0.18
16.5
3.4
4.8
0.08
57
(16)
4.7
1.2
0.087
13.4
3.5
3
0.09
31.2
Table 8. A table showing survival rate and the combined total predator populations at the peak position of predator species survival and the position
with the largest combined predator population size.
Predation
strategy
It can be noted how these points do not correspond, the point at which the most predator species survive is not the same as the point that the ecosystem is most populous.