Space-Discrete Model
Posted: August 25th, 2021
Space-Discrete Model
Name
Institution
Space-Discrete Model
Abstract
Since the formation of exponential growth in 1798 (Malthus), the study of the population has led to the development of mathematical models to describerelated phenomena.Behavioural discoveries like the self-regulation of the population when resources are low and better performance for groups’survival gave rise to logistic growth and Allee effect models, which suggest how individuals can function independently.In this research, the two models are the objects. Concentration would be placed on the definition of equations, the general behaviour of their solutions, the stability of equilibrium solutions, and coupling of populations with a constant intensity. An elaborate discussion would first concentrate on the logistic growth model. The popular logistic growth model is one of the most studied models in demography. Consequently, its applications have inspired many researchers to propose extended versions of the model. In 1932, Warder Allee and Edith Bowen discovered that each specimen of a group of goldfish could survive longer than isolated individuals in an environment that is polluted with colloidal silver. Allee effect model allows an analysis of different situations depending on the intensity coupling of sets of groups. Results from the logistic growth models showed accurately that the natural boundedness that a population has as a response to a limitation in availability of resources, refining the classic exponential growth model. Additionally, overcrowded populations that follow the Allee effect tend to stabilize without extinction when interacting with other species. Hence, the study conclude that models of population growth, which involve polynomials of superior degree, could improve the accuracy that has been obtained with the current ones.
Contents
Introduction
Space-Discrete Model
Logistic Growth Equation
System of the logistic growth equations with coupling between two different habitats
Isoclines of the logistic growth equation
Allee Effect Equation
System of Allee effect equations with coupling between two different habitats
Isoclines of the Allee effect equation
System of Allee effect equations with coupling between different habitats
Conclusions
Bibliography
Introduction
This model, although describing accurately the evolution of a population at the first moments, inhibits usersfrom predicting the long-termbehaviour.Malthus (2015, p. 178) recognizes theissue by asserting that the model does not account for the limitation in resources,andfails to provide a concise solution. However, Slavov et al. (2014) confirm that constant cell growth rate is possible under conditions of decreasing energy flux. In this study, the dynamics of a species is considered as spatially structured, inhabited in two distinct environments that are subject to changes precipitated by individual behaviour, which is referred to as coupling in two unique sets of habitats. Space-discrete models, aided by computer simulations using Matlab software, are employed to help show that it is possible for two regimes exhibiting completely different dynamics to exist. The variations in the interactionsbetween the subpopulations, the aggregate existence is bound to either reduce or rapidly increase.Additionally, the logistic growth model also featuresan inflection point, which is the size of the population that has the maximum growth rate. The change can be interpreted as a rapid increase in the levels of population growth until the inflection point before beginning to decelerate (Tsoularis &Wallace, 2002, p. 26).Decrease in population levels reflects the points of inflection.
The assessment of the two models is divided into two main parts. The first part makes an insight of the logistic growth equation, the general behaviour of its solutions, and the stability of its equilibrium solutions.In this part, the logistic growth equation also plays a role in defining a model of coupling two populations. This study helps to discover steady states and stability to find equilibrium for both populations simultaneously.The second part assesses the Allee effect equation. Away from the components suggested in the first part, this sectionis a comparisonbetween the Allee effect equation and the logistic growth equation. At the end of the second part, examples of systems of Allee effect equations with coupling between several populations are exhibited. To do so, instead of showing each explicit solution or isoclines, time steps and equilibriums are calculated with the help of mathematical software Matlab.
Space-Discrete Model
Mathematical models are developed to describe natural phenomena. Space-discrete modelling is particularly useful because it can be achieved by statistical information, and it has gained acknowledgment due to the development of computing. Study of population dynamics has used this resource since its beginnings: a typical example is a study performed by Pearl and Reed (1920) about the rate of growth of the population of the United States. In this study, the authors considered the data of population provided by the Census Bureau every ten years since 1790 until 1910. Then they use the method of minimum squares to determine a logarithmical model that workedfor over 50 years, obtaining a relative error of less than 1% in the estimatedperiod. They also recognized that this model would not work in later years.For example, the model predicted a population for the United States of 11 billion people for the year 3000 AD, an absurd value.Later, the logistic growth model with the same level of accuracy than the previous logarithmic model was suggested, effectively eliminating the problem of unboundedness that led to unrealistic predictions.
Logistic Growth Equation
The logistic growth model for populations was first proposed by the Belgian mathematician Jean-Pierre François Verhulst in 1838(Vogels et al., 1975) as an alternative to the problem of unboundedness of the exponential growth model by Thomas Malthus.TheMalthus modeldoes not describe the growth of a population after a considerabletime. In the logistic growth model, the growth rate of the population is a second-degree polynomial in whose roots are and and it has a negative leader coefficient. The constant is sometimes called equilibrium populationandcoincides with the carrying capacity(i.e., the maximum population that can be sustained by the environment at any time). In modern terms, the logistic growth equation takes the form
To identify the general behaviour of the solutions of the logistic growth equation (Eq. 1), the parabola should be studied. Observe that for if and only if . Furthermore, for a fixed value , is greater when the value is greater. This means that a solution for equation (Eq. 1) has a higher growth rate when is greater.Hence the constant is sometimes called theintrinsic growth rate of the population .Equilibrium points ( ) of equation (Eq. 1) are and . Furthermore, for every population which satisfies the logistic growth equation (Eq. 1), it is obtained that as . Let be a solution of equation (Eq. 1) such that .Then at , which means that the population is increasing. If for some , it follows that at . Then for every and the statement follows. If for every , thenthere exists a number such that when . If , then while , which contradicts the fact that the polynomial has exactly two roots.Finally, if , then at , which means that the population is decreasing. A similar argument applies.
The benefit of the inflection point of a curve as the point is to indicate the point when the population reaches its maximum increasing (or decreasing) rate. In the case of the logistic growth equation, this inflection point can be obtained again by looking at the parabola . This curve has its maximum value at its vertex, this is, the point . Thenthe value is the inflection point of population growth in the logistic model. In other words, the population grows at the maximum speed when reaches half the carrying capacity.Population growth starts to decelerate past this inflection point.Recall that if for certain time , then and
This means that the population starts to decrease untilit eventually reaches the equilibrium point . The behaviour of this equation in this situation is coherent with the phenomenon of overpopulation such that environmental resources are not enough to let the population grow even more. Figure 1 illustrates the possible situations with carrying capacity and intrinsic growth rate .The yellow curve shows the behaviour of the population if : population grows monotonically to the stable equilibrium. Observe that the measurement started at the inflection point; thus, the population grew at its maximum speed at this initial point. The blue curve shows the behaviour of the population if : population decreases going nearer to the carrying capacity . The red curve shows the behaviour of the population if : the population is at its stable equilibrium.
Figure 1. Logistic growth model ( , ).
System
of the logistic growth equations with coupling between two different habitats. When two species
and
, with
respective carrying capacities
and
and respective intrinsic growth rate
and
,interact
in the same environment, growth of each population also depends on the population
of the other species.
Figure 2. Relationship between two populations in logistic growth
From figure, the existence of species in two different habitats is inter-dependent. Increase in the population of species influences the growth in the population of with coupling between two different habitats.
Assuming the logistic growth for each population separately,the new equations are
The constants and , which are usually considered to be equal, are sometimes called the intensity of coupling (Petrovskii &Li, 2001).In some cases, the intensity of coupling could mean a fraction of the population that is exchanged (Hastings, 1993).Theanalyse of the coefficients and notations from the book,“The Mathematics Behind Biological Invasions” (Lewis, Petrovskii and Potts, 2016, s. 2.2.1.2) will be followed.Consider the functions
Denoting by the derivative of with respect to and by the derivative of with respect to , it is calculated that and . Thenthe two species are mutually beneficial if both and are positive, competing if both and are negative,and one species predates the other if and have different signal: more explicitly, if is positive and is negative, then is the predator and is the prey; also, if is negative and is positive, then is the prey and is the predator.
Isoclines of the logistic growth equation. It is possible to calculate the equilibrium populations in the system formed by equations (Eq. 2) and (Eq. 3). By making , the expression obtained from equation (Eq. 2) is
Similarly, by making , equation (Eq. 3) is transformed into
Equations (Eq. 4) and (Eq. 5) can be plotted in a rectangular coordinate system whose horizontal axis represents the population and whose vertical axis represents the population (Figure 1). In this system, equation (Eq. 4) looks like a vertical parabola (red curve). On the other hand, equation (Eq. 5) looks like a horizontal parabola (blue curve). In the example illustrated in Figure 1, intrinsic growth rates are and , carrying capacities are and , and the coefficients of interaction between the populationsare . The system of equations is;
and the equations of the isoclines are;
The intersections between these two curves represent the momentswhen and ; this is when the two populations reach the equilibrium simultaneously.The calculation of the intersection points leads to resolving of polynomial equations. An alternative is to find approximate solutions by plotting the isoclines that provide the coordinates of the intersection points. In this example, the intersection between the two isoclines corresponds approximately to the point . In other words, the steady-state of the system of logistic growth equations corresponds to the populations and .
Figure 3. Isoclines of logistic growth and equilibrium populations.
To determine the stability of the steady-state , the derivatives of and are calculated:
Then the determinant of the system is(Lewis, Petrovskii and Potts, 2016, equation (2.32))
and the trace is
which means that the steady-state is stable.It is also worth to observe that the species in this example are mutually beneficial as both intensities of coupling are positive.
Allee Effect Equation
In the logistic growth model, the population of the species is estimated by the ideal conditions of the growth depending only on the environmental resources. It is a monotonically growth that gets stable after a long period, going closer to carrying capacity. In 1932, Warder C. Allee and Edith S. Bowen published the results of an experiment, which involved the survival of goldfish in an environment that has been polluted with colloidal silver. Among many conclusions, it was observed that groups of goldfish in such environment managed to survive longer than the isolated fish.
More evidence of Allee effect was found with spawning Pacific salmon during spring of 2012 in the Kenai Peninsula, Alaska (Mooring et al., 2004). Their growth rate increased by 268% while the energy density increased by 175% due to the nutrients transported from the ocean by their progenitors, and this effect persisted by beyond six months (Rinella et al., 2012).
Another evidence of the Allee effect could be found in the desert of New Mexico, where bighorn sheep are constantly in predation risk. It was observed that the biggest males in ram groups were the most exposed to predation as these groups are small when compared with other groups of sheep (Mooring et al., 2004). Similarly, the fear of predators increased by seven times the risk of extinction of less dense groups of Drosophila melanogaster (Elliott, Betini & Norris, 2017). As all evidence suggests, high levels of predation threaten extinction.
If particular menace makes part of the habitat for a considerable time, the behaviour of the species could change according to the number of individuals. If the population is low, the outcome could be extinction. If the community is high (of course, not overpassing the carrying capacity), it could manage to grow. The minimum number of individuals that guarantees the non-extinction of the species is called the critical population and is denoted by .
Equation (Eq. 6) is called the Allee Effect Equation.If the term to the right in equation (Eq. 1) is denoted by and the term to the right in equation (Eq. 6) is denoted by , the following relation is obtained:
This means that the equilibrium points of the Allee effect equation (Eq. 6) are the equilibrium points of the logistic growth equation (Eq. 1), and , and the additional equilibrium point . For example, when , , and , the function looks like in Figure 4:
Figure 4. Graph of vs in Allee effect.
This means that population must decrease when and that it should increase when .
Like in logistic growth, the Allee effect model also presents inflection points between equilibrium points. It is straightforward to determine inflection points when a graph like Figure 3 is provided: it is just necessary to look at the local extrema of the cubic curve. These inflection points can also be calculated by differentiating as:
The zeroes of the polynomial are the inflection points, this is,
and
Figure 4 shows the behaviour of a variety of populations, which satisfy the same Allee effect equation. The carrying capacity is and the critical population is . Substituting these values in equations (Eq. 7) and (Eq. 8), the inflection points are
and
These inflection points again indicate that the greatest increasing rate is at and that the greatest decreasing rate is at .The green curve corresponds to aninitial population (inferior to the first inflection point): the result is that the population starts to decrease very fast, then stabilizes to zero. The purple curve corresponds to another initial population , this is, the initial population between the first inflection point and : the result is an initial slow-paced extinction, going faster when reaching the inflection point. The yellow curve corresponds to an initial population , which is between, and the second inflection point: the result is an initial slow-paced growth, going faster when reaching the inflection point. The red curve corresponds to an initial population , near to the second inflection point: the result is an initial fast growth with a subsequent stabilization to the carrying capacity . The dark blue curve corresponds to an initial population : the result is a population decreasing to the carrying capacity. The light blue curve corresponds to the initial population : the population remains constant as it is the equilibrium point.
Figure 5. Allee effect model ( , ).
The behaviour of the solutions with initial value near shows that the solution is an unstable equilibrium.In fact, if for some , then for every ; if for some , then decreases for every , and if for some , then increases for every (Keitt, Lewis &Holt, 2001, Figure 1, Bottom).
System of Allee effect equations with coupling between two different habitats. Interaction between species can also be modelled when it is assumed that they follow the Allee effect when isolated.
The
following graph illustrates a typical solution of the system that is formed by
the equations (Eq. 9) and (Eq. 10). The example depicts mutualism between two
populations
and
with initial values
and
.
Figure 5. Typical solutions of a system of Allee effect equations.
Figure 6
Depending on the intensity of coupling and the initial populations, the equilibrium could be reached at positive values, as in the example, or at the extinction of both species. According to Figure 5, it is shown that there is a wide variation in the population of the two species. However, after recovery of species , species also recovers when and with the extinction of , the predator, also goes into extinction simultaneously. For the following examples, two species and are considered with carrying capacity and respectively. They have initial values and . It is first considered an intensity of coupling , which leads to the following behaviour;
Figure 7a). Solutions of a system of Allee effect equations, .
The least intensity of coupling makes each population evolve as in the original Allee effect model (Eq. 6). After increasing the intensity to , equilibrium is reached, which favours :
Figure 7b). Solutions of a system of Allee effect equations, .
With an intensity of coupling , population getseven more prejudiced. In other words, the habitat becomes increasingly unfavourable, but the equilibrium of remains the same as before. Hence, it means conditions for survival are becoming less useful.
Figure 7c. Solutions of a system of Allee effect equations, .
With the high intensity of coupling , the situation gets unsustainable for population , leading to a fast extinction. Population also fades at a slower pace:
Figure 7d. Solutions of a system of Allee effect equations, .
From figure 9, the interaction between the two species occurs at the time, t = 45 when the population of the two species is equal, u1 = u2. However, it is observed that the smaller population gets benefited with some of the interaction with the other species. However, if this interaction grows too much, the larger population could go extinct very fast, leading to the extinction of the small population. An explanation of this phenomenon could be found in study of isoclines, which shall be done in the following section.
In summary, an increase in coupling intensity is unfavourable particularly for one species as it is likely to undergo faster extinction as compared to the other. However, there is wide variation in the population of the two species when value . However, with increase coupling, q = 0.8, the variation decreases as the two species goes into extinction. Equilibrium in the inhabitant is obtained when the equality, arise after which the extinction of species of moves faster than that of species .
Isoclines of the Allee effect equation. Like it was made with equations (Eq. 2) and (Eq. 3), isoclines of the Allee effect can be determined by making in equation (Eq. 9) and in equation (Eq. 10) to mean that member 1 (Eq.9) is attaining a different shape as compared to the second part of Eq10, hence resulting relation between populations is
while in the second situation, the resulting relation is
Again, the graphs of these two equations can be plotted in a rectangular coordinate system (Figures 10 to 12). The red curves correspond to equations of the form (Eq. 11), while the blue curves correspond to equations of the form (Eq. 12). The values for , , , , , and are held constant in the three examples and the value varies in the set . The Allee effect equation system is then
and the isoclines are
The isoclines intersect at two different points when , meaning that there are two steady states with this intensity of coupling:
Figure 8a). Isoclines of Allee effect, .
As in the case of logistic growth equations system, steady states are determined with the aid of a graphing calculator. These steady states are approximately and .
When , the only steady-state is the trivial one :
Figure 8b). Isoclines of Allee effect, .
From figure 8a, in a spatially structured environment, as the interplay intensity between the subpopulations increases, aggregate abundance is likely to decrease or experience abrupt out-burst in population size. The continuous model used in figure 1 shows that both ‘self-inhibition’ and ‘self-excitation’ regimes in the population systems are strongly subjective to the changes in the size of the habitat(Petrovskii &Li,2001, p. 549-562). However, if the population of one regime is larger than the other, there are chances that ‘self-excitation’ would yield to way for ‘self-inhabitation’, the reverse is also true.
Similarly, the only steady-state for is the trivial one .
Figure 8c). Isoclines of Allee effect, .
Indeed, equation (Eq. 11) can be rewritten as
Similarly, the expansion of equation (Eq. 12) gives rise to the equation
Then the coefficients and act as scaling concerning the line (Young, n.d.). When gets bigger, these coefficients get smaller and this makes the isoclines get compressed on the curve (the black diagonal line in all the three graphs). For comparison, observe how the blue isocline in Figure 10a has very pronounced curves (small ) as opposed to the very flat shape in Figure 10c (large ). This operation of flattening the curves does not need to preserve the number of intersections between the two isoclines. As an additional observation, the black dots in each figure, which are the intersections of the isoclines with the curve , have coordinates , and , and they are fixed points in all the isoclines when just the intensity of coupling varies.
In order to determine the stability of the steady states that were found, it is proceeded as in the case of logistic growth equation system. Like before, denote;
Then;
The derivatives of these functions are
When and , the values of these derivatives are
Given that;
,
Then the determinant of the system is
which means that the steady state is unstable. Now if , the values of the derivatives are
Given that;
But;
Then the determinant of the system is
and the trace is
which means that the steady state is stable.
System of Allee effect equations with coupling between different habitats. Now it is supposed the existence of one hundred of species , …, with their respective carrying capacity , their respective intrinsic growth rate and their respective critical population in the same environment. It is also supposed that the population only interacts with the populations and . It is formed a system of 98 Allee effect equations
For example, to see how the carrying capacity and the critical population affect the evolution of a population with a fixed initial value, it could be required that , which corresponds to the equation of the red line in Figure 13, and , which corresponds to the equation of the yellow line to associate. The value of could be equal and positive in all the equations of the system every time and the initial value of each population would be 5 (dashed red line). In other words, it is assumed that all the populations have initially the same number of individuals, interact with other populations with the same mutualism intensity coupling, but every population has its carrying capacity and critical population, and they have been ordered ascendingly. Numerical equilibrium solutions (blue circles) of the system are calculated with mathematical software like Matlab. In the case when , the behaviour in Figure 9a is obtained:
Figure 9
Figure 9a. Equilibrium solutions for a 98 Allee effect equation system ( ).
According to Figure 9, populations that have a critical value that is smaller than the initial population go to a stable equilibrium equal to their carrying capacity. On the other hand, populations that have a critical value that is greater than the initial population go extinct. This behaviour is expected since the intensity of coupling is low. Hence the interaction between populations is little when compared with the part of the equation that corresponds to the Allee effect. Then each family evolves according to the original Allee effect equation (Eq. 6). Recall that in this model, populations that have an initial value below the critical population go extinct, and populations whose initial value is above the critical value tend to the carrying capacity.
Changing the intensity of coupling to would derive in the situation of Figure 14:
Figure 9b. Equilibrium solutions for a 98 Allee effect equation system ( ).
Figure 9b shows that the populations with a carrying capacity below the starting value, and just a few above it have positive stable equilibrium, and most of these populations tend to their carrying capacity. The remaining populations go extinct.
Changing the intensity once more to , equilibrium populations look as in Figure 9c:
Figure 9c. Equilibrium solutions for a 98 Allee effect equation system ( ).
With this high intensity, only populations with low carrying capacity have a positive stable equilibrium. Populations with carrying capacity greater than the initial population goes extinct. By fixing the time , a complete set of values can be plotted, providing a time step. Figure 10 shows some time steps for , , , , and .
Figure 10. Time step for a 98 Allee effect equation system ( ).
Recall that the time step at is represented with the dashed red line. Evolution of populations is perceived through different time steps. The time steps at and look like Figure 9. This means that the Allee effect equation system has reached a steady-state different from the trivial .
Figure 11
Now, parameters and are changed by random values with the Matlab function rand as the first approach to stochasticity (Lidicker Jr., 2010). The result is depicted in Figure 12.
Figure 12. Random parameter values for (blue circles) and (red circles).
With these parameters and the initial populations , it is obtained that all the populations ultimately go extinct.
Figure 13. Equilibrium solutions, random parameters.
As before, time step helps to figure out when equilibrium is reached. Since many critical points are above the initial population and the intensity of coupling is high ( ), it is expected that extinction would occur very early as it is seen in Figure 14.
Figure 14. Time step, random parameters.
Like in the previous example, there is an evolution of populations between time steps. The steady-state occurs at early time .
By changing parameters again to , , , and (i.e., no interaction between populations), it is obtained an Allee effect system where all its equation are original Allee effect equations (Eq. 6 ) to represent changing away from the limits. With these parameters, it follows that the first 50 populations go extinct and the remaining grow to the carrying capacity. Figure 20 shows these equilibrium states.
Figure 15. Equilibrium states for a 98 Allee effect equation system.
As shown in figure 20, the invasion occurs on a 2-dimensional lattice under which the initial occupancy conditions are spatially and randomly distributed. At the critical point of initial occupancy level, q = 0.01 there is no invasion and these initial occupancy levels persists indefinitely. However, above q = 0.01 when population becomes randomly seeded, invasion occurs and population occupies the whole lattice.The behaviour is expected since in Allee effect model (Eq. 6), populations that have initial value above critical value, for instance, assuming a value q=0.1 or q = 10,there would be a change and movement in population that tends to their carrying capacity, and populations that have an initial value below critical value go extinct.
Conclusions
Different factors influence the existence and relationship between the population and the environment as well as its interaction with other population species within the environment. As such, different models exist to help understand the behaviour of the population over time. These behaviours vary from growth and survival while interacting with others. Population growth can be modelled in many ways depending on the factors that are considered. The study used logistic and Allee growth models to help understand the growth behaviour of a population when it interacts with others. The mathematical variation in the two models was discussed concerning their approach and limitations when modelling population growth. Logistic growth is one of the simplest models for this achievement and it shows accurately the natural boundedness that a population has as a response to a limitation in availability of resources, refining the classic exponential growth model. The model shows that the population of a particular species gets to extinction when the carrying capacity reaches the maximum.Allee effect has been observed in the last century as a paradigm of a factor that can drastically change the behaviour of a population beyond the logistic growth model. The examples that were studied showed that overcrowded populations that follow the Allee effect tend to stabilize without extinction when interacting with other species. However, a very high interaction could be detrimental to both populations, leading to extinction. Furthermore, the Allee effect with coupling can present both stable and unstable steady states in the same system (Figure 10) or only the trivial steady state (Figures 11 and 12). Both situations are impossible in the logistic growth with coupling: there is always exactly one steady-state different from the trivial one (Figure 2). In this fashion, the Allee effect model presents more variety of situations than the logistic growth model. Allee effect with coupling and stochastic parameters (Figures 17 to 19) is of special importance to analyse and to improve as it represents the discrete individual behaviour of each population better than the continuous parameters (Stephens, Sutherland and Freckleton, 1999). Models of population growth, which involve polynomials of superior degree, could improve the accuracy that has been obtained with the current ones. For example, a model with an equation defined by the right-hand term of the Allee effect equation (Eq. 6) multiplied by a linear factor, given a result of polynomial degree 4 with all the variations that this mathematical object could provide (Dunham, 1991, Figure 1). However, further observation in nature is needed to find any actual significance for such factor.
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