Numerical Investigation and Factor Analysis of Two-Species
Spatial-Temporal Competition System after Catastrophic Events
YOUWEN WANG1, MARIA VASILYEVA1, SERGEI STEPANOV2, ALEXEY SADOVSKI1
1Department of Mathematics and Statistics,
Texas A&M University Corpus Christi,
Corpus Christi, Texas,
UNITED STATES OF AMERICA
2Institute of Mathematics and Information Science,
NorthEastern Federal University,
Yakutsk, Republic of Sakha (Yakutia),
RUSSIA
Abstract: - The interaction of species in an ecological community can be described by coupled system partial
differential equations. To analyze the problem numerically, we construct a discrete system using finite volume
approximation by space with semi-implicit time approximation to decouple a system. We first simulate the
converges of the system to the final equilibrium state for given parameters (reproductive rate, competition rate,
and diffusion rate), boundaries, and initial conditions of population density. Then, we apply catastrophic events
on a given geographic position with given catastrophic sizes to calculate the restoration time and final
population densities for the system. After that, we investigate the impact of the parameters on the equilibrium
population density and restoration time after catastrophe by gradually releasing the hold of different
parameters. Finally, we generate data sets by solutions of a two-species competition model with random
parameters and perform factor analysis to determine the main factors that affect the restoration time and final
population density after catastrophic events.
Key-Words: - Multispecies competition model, Numerical investigation, Factor analysis, Catastrophic event,
Population dynamics, Ecosystem, Spatial-Temporal model, Lotka–Volterra model, Finite
difference approximation
Received: July 24, 2022. Revised: March 18, 2023. Accepted: April 9, 2023. Published: May 16, 2023.
1 Introduction
Natural and artificial catastrophes disturb human
and natural environments, [1], and cast impacts on
species. For instance, unrestrained hunting in Sabah
(Malaysia) between 1930 and 1950 caused a drastic
population decline of the Sumatran rhino, [2], an
outbreak of yellow fever in Argentina between 2007
and 2009 threatened endangered brown howler
monkey populations, [3]. In the marine
environment, human-caused oil spills can have
devastating ecological effects, as evidenced after the
Ixtoc blowout in the Gulf of Mexico during 1979-
1980, zooplankton decreased in biomass levels by
almost four orders of magnitude more than observed
in 1972, [4]. This work focuses on studying
predictive factors for species restoration time after
the catastrophe event. Finding factors impacting
restoration time can provide better conservation
decisions and minimize recovery time, [5]. The
following are some factors that can affect species’
populations after catastrophes according to previous
studies: species life-history strategies (i.e., the
tradeoff between growth, survival, and reproduction.
For example, fast-lived species are better than slow-
lived species in terms of recovering after climate or
land-use change), spatial area of human-assigned
natural reserves, communities’ political, social, and
financial capitals, the age distribution of species,
dispersal distance, connectivity, catastrophic
mortality, initial population size, environmental
stochasticity, demographic stochasticity, density,
sex ratio, harvest, genetic variation, etc., [6], [7],
[8], [9], [10], [11], [12], [13], [14], [15], [16], [17],
[18], [19].
In this paper, we study the key factors that
influence population dynamics. With the
improvement of computational power, more factors
can be included in the prediction model. However,
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the increased number of factors creates complexity
and difficulty in the simulation of the species
restoration process. Furthermore, we also need to
have methods that allow us to identify which factors
are the most important in species recovery so that
we can allocate our conservation resources and
minimize costs. This search for key factors has been
intensively studied via the combination of field data,
[20], and simulation techniques such as population
viability analysis (PVAs), [21]. The PVAs include
various key habitat factors to predict the population
dynamic and risk of extinction of species using
mathematical models, [22]. The PVA approach has
been a core methodology in conservation science
over the last three decades. It can utilize at least
three types of models, [23]: (1) simple occupancy
models for metapopulation, which are parameterized
using data on the presence or absence of a species in
habitat patches but ignoring demographic data (sex,
age, stage, etc.); (2) structured population models,
which incorporate the spatial structure of habitat
patch and species’ internal dynamic (age structure,
immigration, density, etc.), [24]; (3) most complex
individual-based population models, in which
individual dispersal, survival, and reproduction vary
with respect to their demographic characteristics,
[25], [26]. Multiple PVA packages can serve the
simulation purpose. For example, the ZooRisk
package supports faster analysis of ex-situ
populations, while the VORTEX package can be
used when the data, expertise, and time is adequate
to explore complex individual-based metapopulation
models, [27]. After PVA simulation using data of
species, sensitivity analysis is applied to determine
the key factors that affect species' survival, [28],
[29], [30], [31], [32]. However, there are some
criticisms of the PVA approach, for instance,
significant differences were noticed in terms of
prediction by different PVA packages, [33],
although catastrophe is verified to have a strong
effect on PVA outcome, the proportion of studies
that examined this effects did not increase over
time, [34], additionally, PVA is effective for
evaluating the relative extinction risks of different
species, but it shouldn’t be used to estimate the
likelihood that a certain species would become
extinct, [35].
In this work, instead of using the PVA approach
considering multiple habitat factors, we simulate
multispecies competition based on the Lotka–
Volterra model, which is used to describe the
population dynamics of species competing for some
common resource, [36]. We also combine the
multispecies model with the simulation of the effect
of catastrophe. In this way, we can study the
dominant factors of species recovery after the
catastrophe event. Population viability analysis
(PVA) and Lotka–Volterra multispecies competition
model are methods for simulating population
dynamics, but they differ in their goals,
assumptions, and complexity. PVA is designed to
predict population persistence or extinction under
different scenarios. In contrast, the Lotka–Volterra
model is designed to simulate species interactions
and the potential for extinction due to competition.
The Lotka-Volterra competition model uses an
interaction matrix to describe the dynamics of
multiple species interacting pairwise. It has been
used in many areas: Industry Competition, Genetic
Drift, Ecology, Epidemiology, Game Theory,
Sociology, etc., [37], [38], [39], [40], [41]. In the
Population Dynamic of species, this model has been
intensively used to study the impact of the shift of
environment, [42], [43], [44], [45], [46], [47], [48],
[49], [50]. It’s a powerful tool for studying the
dynamic of species after the catastrophe in that it
can model population recovery, [51], the connection
between climate feedback and mass extinction under
the competition for limited resources, [52], the
connection between spatial heterogeneity and
robustness of ecosystem after catastrophe, [53],
feedback loops, [54], etc.
In the simulation result analysis, statistical
techniques such as factor analysis and sensitivity
analysis are used to identify the main factors that
affect the restoration time or equilibrium population
after the catastrophe. However, they differ in
purpose and approach. While sensitivity analysis is
used to identify the most important input variables
that affect the output or response of a particular
model or system, [55], factor analysis is used to
identify underlying factors that explain the variation
in a set of measured variables, [56]. Note that
sensitivity analysis is widely used in analyzing
ecosystem datasets, but applying factor analysis on a
nonlinear system is rarely studied. Therefore, in this
work, we first numerically investigated the post-
catastrophe ecosystem from a perspective of species
competition, then we used numerical simulation to
generate a dataset with random values of different
factors, at last, we Therefore, this work applies
factor analysis to simulated nonlinear system
datasets and interprets the simulated dataset in a
new way. In this paper, we employed a four-stage
methodology to investigate the dynamics of a two-
species competition model and the impact of
catastrophic events on system recovery. First, we
simulated the convergence of the system to its final
equilibrium state using given parameters,
boundaries, and initial population densities. Next,
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we introduced catastrophic events at specific
locations with specific geographic sizes and
calculated restoration time and final population
densities. We then analyzed the effect of model
parameters on equilibrium population density and
restoration time by gradually releasing their hold.
Lastly, we generated data sets using random values
of factors and performed factor analysis to identify
key factors influencing restoration time and final
population density after catastrophes.
The paper is organized as follows. Section 2
describes our problem formulation, introduces the
mathematical model used for simulation, and
presents numerical results with some fixed sets of
parameters. In Section 4, we make factor analyses of
simulated datasets. Section 6 concludes the work
and discusses future works.
2 Problem Formation
We consider a two-species competition model in
one-dimensional domains Ω = [0, 1]. The
mathematical model is described by the following
coupled system of equations, [57], [58]:
( 1 )
with some given initial condition
and fixed boundary conditions for both species,
Here and are the population of the
first and second species, and are the diffusion
coefficient, and are the first and second
species reproductive growth rate, and are
the interaction coefficient due to competition.
We define a uniform mesh:
where is a positive integer and . Let
be a time step, and for . For
numerical solution, we use a finite difference
approximation by space with a semi-implicit scheme
for time approximation, Then, for
and
, we obtain the following
discrete form
with . ( 2 )
To simulate pre and post catastrophic cases, we
have the following algorithm:
Pre-catastrophic case: Solve the system (2)
with some given constant initial condition:
to find an equilibrium state ( and
) and the time needed to reach it
().
Post-catastrophic case: We use the previous
(pre-catastrophic) solution and apply
catastrophic events in some subdomain
Then solve the system (2) with the initial
condition:
until the system reaches an equilibrium state
( and ) and records
the restoration time ( ). When the
change in population density is less than
, the equilibrium is considered
reached.
Next, we performed numerical simulations based
on the presented algorithm. Before we release
control of values of all parameters during
simulation, we first controlled the value of all
parameters (reproductive rate, competition rate,
diffusion rate, boundaries, initial conditions of
population density, and catastrophic size) in section
2.1.
We consider two cases:
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Case 1 (one species survive)
Case 2 (both species survive)
with regular diffusion and small diffusion
. In simulations, we used a grid with
nodes and performed simulations with
with initial conditions for
pre-catastrophic cases.
After that, we gradually released the control of
catastrophe size (section 2.2) and diffusion rate
(section 2.3) to understand the dynamic of the
system. Finally, we released control of all
parameters and simulated catastrophic events
(section 3).
2.1 Solution and Dynamic for Two-Species
Competing Model Pre- and Post-Catastrophe
2.1.1 Case 1 (One Species Survives)
Figure 1 presents the case in which only one species
survives when we control all parameters. We
observed that, in comparing regular diffusion with
small diffusion, the latter takes more time to reach
equilibrium, both before and after the catastrophe.
Specifically, species with small diffusion take
876-time steps to reach equilibrium before the
catastrophe, whereas species with regular diffusion
take only 269-time steps. After the catastrophe,
species with small diffusion take 2689 − 876 =
1813-time steps to reach equilibrium, while species
with regular diffusion take only 637 − 269 = 368-
time steps, which is five times faster.
Furthermore, the boundary constraint has less
effect on small diffusion. If diffusion is small, at the
final equilibrium state, the central highly populated
area is larger than with regular diffusion. At
equilibrium, around 40% of the central geographic
domain has a population density above 0.8 with
regular diffusion, while around 80% has a
population density above 0.8 with small diffusion
for surviving species.
2.1.2 Case 2 (Two Species Survive)
We also examined the case in which both species
survive, as is shown in Figure 2. We observed that
small diffusion leads to a shorter time to reach
equilibrium compared to regular diffusion, both
before and after the catastrophe. This situation is the
opposite of what we observed in case 1, where
species with small diffusion take more time to reach
equilibrium.
Specifically, before the catastrophe, species with
small diffusion reached equilibrium in 226-time
steps, slightly faster than species with regular
diffusion, which took 276-time steps. After the
catastrophe, species with small diffusion reach
equilibrium in 488 − 314 = 174-time steps, which is
twice as fast as species with regular diffusion,
reaching equilibrium in 637 − 276 = 361-time steps.
Same as in case 1, the boundary constraint has
less effect on small diffusion. At equilibrium,
around 20% - 50% of the central geographic domain
has a population density above 0.5 with regular
diffusion, while around 80% - 85% has a population
density above 0.5 with small diffusion for surviving
species.
2.2 Effect of the Catastrophic Size
As is shown in Figure 3, we controlled all other
parameters and only released the control of
catastrophe size. We found that as the catastrophe
size becomes larger, the restoration time slightly
increases, both in case 1, and case 2.
Specifically, we observed that under case 1,
when we place a catastrophe to the system after the
system has reached its pre-catastrophe equilibrium
(at time step 269), the number of time steps to reach
equilibrium after a catastrophe is 368, no matter
what the size of the catastrophe is (5, 25, 50, 75%).
Under case 2, when we place a catastrophe to the
system after the system has reached its pre-
catastrophe equilibrium (at time step 276), the time
steps for the system to reach equilibrium slightly
increase from 345 (when the size of the catastrophe
is 5) to 387 (when the size of the catastrophe is 75).
2.3 Effect of the Diffusion
As is shown in Figure 4, we controlled all
parameters and only released control of diffusion,
and we observed that diffusion is the key factor for
determining the survival status groups.
Specifically, similar diffusion rate combinations
before and after the catastrophe lead to similar
survival statuses for both species. Borderline
combinations require more time steps to reach
equilibrium. After the catastrophe, diffusion rates
still determine survival status, with little change
except for borderline combinations.
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3 Factor Analysis for Random
Parameters
We performed 100k simulations with random input
parameters, with the following scale for
reproduction rate, competition rate, diffusion rate,
and initial condition of population density.
Next, we take a simulation that leads to the cases
where at least one species survives (73k). Table 1
presents the proportion of the survival group before
the catastrophe stroke. In more than 80 % of the
scenario, only one species survives. Finally, we
apply catastrophic events with random length:
.
Table 1.Proportion of the survival group before the
catastrophe stroke. In more than 80 % of the
scenario, only one species survives.
Regular diffusion,
Small diffusion,
Fig. 1: Case 1: One species survives. Solutions at the final time in the first and second columns, the
solution average over the domain versus time in the third column, under regular diffusion with
(first row) and small diffusion with (second row). Solid lines represent before the
catastrophe, while dashed lines represent after the catastrophe with a size of the catastrophe
. The red line represents species 1, and the blue line represents species 2.
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Regular diffusion,
Small diffusion,
Fig. 2: Case 2: Both species survive. Solutions at the final time in the first and second columns, the
solution average over the domain versus time in the third column, under regular diffusion with
(first row) and small diffusion with (second row). Solid lines represent before the
catastrophe, while dashed lines represent after the catastrophe with the size of the catastrophe
. The red line represents species 1, and the blue line represents species 2.
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Case 1
Case 2
Fig. 3: Restoration time for different catastrophic sizes under regular diffusion
(which means it has the same scale as other parameters such as birth rate and competition rate). Case 1
(only one species survives) is presented in the first row, and case 2 (both species survive) is presented
in the second row.
Pre Catastrophe
Post Catastrophe
Pre Catastrophe
Post Catastrophe
(a) Case 1
(b) Case 2
Fig. 4: Each scatter plot compares and diffusion rates for two species. The size of the
catastrophe is set at . Column 1 and 3 is pre-catastrophe, while columns 2 and 4 show
post-catastrophe with different stopping thresholds (for example. Row 1 shows the time to reach
equilibrium, while rows 2 show the final survival groups and changed survival groups after the
catastrophe: 00 (grey), 01 (blue), 10 (red), and 11 (green). Here 00 means no species survive, and 01
means only the second species survive. In Row 2, the red dots on the boundaries of groups show the
change in survival groups after the catastrophe.
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3.1 Mean and Standard Deviation of Time
Until Equilibrium by Categories
Table 2 and Figure 5 show the mean and standard
deviation of time steps until equilibrium after
categories.
From Table 2, we observed that the restoration
time varies with catastrophe size, with larger
catastrophes leading to longer restoration times and
higher variability.
From Figure 5, we observed that when the
catastrophe size is larger than 0.3, the variation of
restoration time experiences a surge.
In summary, restoration time varies with
catastrophe size, with larger catastrophes leading to
longer restoration times and higher variability,
especially when catastrophe size is larger than 0.3.
This indicates that it is more difficult to predict the
restoration time as the catastrophe size increases.
3.2 Mean and Standard Deviation of
Equilibrium Population Density Solution
Differences pre- and post-catastrophe
Table 3 and Figure 6 show the mean and standard
deviation of solution differences before and after
categories.
In Table 3, we observed that mean solution
differences are consistent across catastrophic sizes,
but standard deviation and maximum values vary
greatly. Catastrophe sizes between 0.3 and 0.4 lead
to the highest standard deviation, making it difficult
to predict equilibrium population density in specific
scenarios.
In Figure 6, we observed that there is an extreme
outlier in the interval [0.3,0.4]. This might account
for the high standard deviation in this catastrophic
group. Yet we can still see that when catastrophe
size is larger, the variation of solution difference is
larger, hence harder to predict.
In summary, the mean solution differences are
consistent across catastrophic sizes, but standard
deviation and maximum values vary greatly.
Catastrophe sizes between 0.3 and 0.4 lead to the
highest standard deviation, making it difficult to
predict equilibrium population density in specific
scenarios.
3.3 Regrouping
Table 4 shows the percentage of regrouping of
survival status after the catastrophe. Survival group
changes occur at a consistently low rate of 2.5-3%.
The probability of regrouping is highest in
catastrophe sizes between 0.3-0.4 and 0.5-0.6,
making predicting species’ survival status
challenging.
Table 2. Mean and standard deviation of time steps
until equilibrium after categories. N = number of
simulations
Table 3. Box plots of equilibrium population density
solution differences of pre vs. post catastrophe
( ),
under different catastrophe sizes
Fig. 5: Post Catastrophe Restoration time steps, under
different catastrophe sizes
Table 4. Post Catastrophe Percentage of regrouping of
survival status
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Fig. 7: Factor Analysis of restoration time steps in two species systems. Dominant (threshold: corr > 0.7)
parameters in each factor are highlighted in yellow. The number of factors is determined by the number of
eigenvalues greater than 1. Left: case 1. Right: case 2.
Fig. 8: Factor Analysis of final population density in two species systems after the catastrophe. Domi- nant
(threshold: corr > 0.7) parameters in each factor is highlighted in yellow. The number of factors are
determined by the number of eigenvalues greater than 1. Left: case 1. Right: case 2.
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3.4 Factor Analysis of Time Steps in Two
Species System
Factor analysis is used to reveal any latent variables
that cause the manifest variables to covary and can
help us to see the trend driving the system, [59]. A
survey of over 1700 PsycINFO studies, including
Factor Analysis, suggested that over 50% of
surveyed researchers used Varimax rotation and
decided the number of factors to be retained for
rotation by Kaiser criterion (all factors with
eigenvalues greater than one), [60]. In this case, the
observed variables are the various factors that
contribute to the species’ restoring time steps.
Factor analysis can help identify the most important
factors driving the variation in the observed
variables.
Figure 7 shows the Factor Analysis of restoration
time steps in two species systems. In both survival
cases, the reproduction rate, diffusion rate, and the
equilibrium population density before the
catastrophe are the most dominant factors. It can be
indicated that to impact the restoration time of
species in the aftermath of a catastrophe, it is
important to examine the diffusion rate and the level
of species population density before the catastrophe.
However, other dominant factors differ between the
two cases. In case 1, the reproductive rate of
species 1 is more important than the diffusion and
equilibrium population density of species 1.
Additionally, competition efficiency is not among
the dominant factors in case 1. This suggests that the
underlying mechanisms driving the species’
restoration time steps may differ in each survival
case. When both species are to survive together,
competition efficiency matters.
Case 1 top factors:
Diffusion of species 2
Reproduction of species 2
Diffusion of species 1
Pre-catastrophe population of species 1
Restoration time
Case 2 top factors:
Diffusion of species 2 and Pre-catastrophe
population of species 2
Pre catastrophe population of species 1
Reproduction of species 1
Reproduction of species 2
Competition Efficiency of species 1
Competition Efficiency of species 2
3.5 Factor Analysis of Final Population
Density in Two Species System
Figure 8 shows the Factor Analysis of the final
population density in two species systems. We
observed that there is a difference in the dominant
factors between the two survival cases. While in
both cases, the most important driving factor is the
pre- and post-population density, followed by
diffusion and reproductive rates, competition
efficiency is not among the dominant factors in case
1, whereas it is a dominant factor for both species in
case 2. It can be indicated that to ensure the survival
of both species in the aftermath of a catastrophe, it
is important to examine the competition efficiency.
Case 1 top factors:
Pre and Post catastrophe population of
species 1
Pre and Post catastrophe population of
species 2
Diffusion of species 2
Reproduction of species 1
Case 2 top factors:
Pre and Post catastrophe population of
species 2
Pre and Post catastrophe population of
species 1
Reproduction and Diffusion of species 1
Reproduction and Diffusion of species 2
Competition Efficiency of species 1
Competition Efficiency of species 2
4 Conclusions
In our simulation study, we numerically investigated
the impact of various parameters (reproductive rate,
competition rate, and diffusion rate) on the
restoration time and final population densities of a
two-species competition model after catastrophic
events. This research holds positive repercussions
for the scientific and academic communities, as it
not only enhances understanding of the post-
catastrophe driving factors of species survival and
recovery dynamics but also presents a methodology
of applying factor analysis to ecosystem restoration
process analysis, instead of applying the sensitivity
analysis.
We first compared the time dynamic and final
population density solutions between two survival
cases (case 1: only one species survives; case 2:
both species survive) under regular and small
diffusion rates. We found that the restoration time is
different for the two survival statuses. For case 1, it
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takes more time to reach equilibrium when both
species have small diffusion. For case 2, it takes
more time to reach equilibrium when both species
have regular diffusion. We also observed that
boundary constraints have less effect on small
diffusion for both survival statuses.
We then investigated the impact of catastrophic
event size and diffusion rate on the restoration time
and final population densities. We found that as the
catastrophic event size increases (especially when
greater than 0.3), the restoration time and final
population density do not change much, but the
variation increases, making it potentially harder to
predict. The diffusion rate is the key factor for
determining the survival status group. Similar
diffusion rate combinations before and after the
catastrophe lead to similar survival statuses for both
species. After the catastrophe, diffusion rates still
determine survival status, with little change except
for borderline combinations.
Finally, we performed factor analysis on the
restoration time and final population density data
sets generated by solutions of a two-species
competition model with random values of
parameters. We observed that for different survival
statuses, the dominant factors and the order of the
factors are different. The dominant factor is usually
reproduction rate, diffusion, and population density
before the catastrophe. However, competition
efficiency is an important factor to consider if both
species are to survive together (case 2), while it is
not the main factor under case 1. This observation
suggested that if our goal is to have both species
survive together, we need to pay attention to the
competitive rates.
In future works, we will study more about the
modeling of catastrophic events. In the real world,
events such as hurricanes, oil spills, disease
outbreaks, hypoxic events, harmful algal blooms,
and coral bleaching all can cause massive species
mortality, [7], [61], however, their simulation may
differ due to variations in spatial patterns. We plan
to randomize catastrophe locations within the spatial
domain, rather than keeping them centralized.
Additionally, we will model various catastrophe
scenarios, accounting for differing species mortality
rates. Lastly, as our current study indicates that
predicting equilibrium population density and
restoration time is more challenging for catastrophes
of larger size, we will employ deep neural networks
to forecast the final state and recovery time of
competing species systems, [62], [63].
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