Stability of Ecosystem: Global Properties of a General Predator-Prey Model - Essay Example

Stability of Ecosystem: Global Properties of a General Predator-Prey Model The purpose of this paper is to offer a summary of the prey-predator model that was established for an attempt to find out how stability could be established with an ecosystem. For a very long time, it has not been easy for a specific model that could accommodate and promote prey-predator satisfactory co-existence. As such, mathematical biology, has introduced several models with an aim of establishing not only a stable ecosystem, but also an atmosphere that could allow organisms that depend on one another symbiotically live together in harmony. One of the models that have been used has not offered consistent results because of its inability to offer specific response. Owing to this, there have been much irregularity and unpredictability of the response results both at present as well as in the future research. In SIR and SIRS models, there is variation in two species whereby one is considered a predator (supplier) while the other one is considered to be a prey (consumer). This dynamism of species relationship form what is commonly referred to as prey-predator type of relationship. In order to resolve unspecific functional responses resulting from the SIR and SIRS epidemic models, an assumption is made and approved by Lyapunov method thereby enabling development of a globally asymptotic stable ecosystem. Introduction There is much complexity in biological processes because of their dependency on factors whose degree of influence is not known. For instance, various factors influencing biological processes can have different effects even in an ecological system with the same properties where the influence is expected to be uniform in all organisms. In such situation, the only way organisms Prey-predator model can produce specific results for ease prediction and understanding of species functional responses is by controlling the response or the behaviour of the system. A typical predator-prey model can be represented as (1, 1): - x (t) = ?(x) ? ?(x, y), y (t) = ? ? (x, y) ? ?(y). The above equation is based on an assumption that in any given ecosystem, there exist an interrelationship between the prey and the predator. Where: - x (t) represents the total population for the prey and y (t) represent the total population of predator. Such interrelationship also imply that the ability of the predator to remain and never be extinct in such ecosystem largely depend on the availability of the prey for consumption. Based on the equation (1, 1), if by any chance the predators are eliminated within the ecosystem, the prey would have a higher probability of producing and growing at a certain rate which is represented by ?(x). That is, ?(x) is the number representing the total growth rate for the pray in the absence of predators. On the other hand, it is expected that number of predators would decrease at the rate of ?(y) especially when the prey are eliminated from the ecosystem. There is non-linear representation in the predator-prey interrelationship. That is, while the prey is expected to increase in the absence of the predators, the reverse is expected on the predators in the absence of the pray. This is attributed to one, the predator depend on the pray for survival and would have limited resource for survival in an event that they prey become extinct in the ecosystem. Secondly, the growth of the prey occurs because of their increased chances for survival in the absence of the predator that is considered an enemy to the predator. On the other hand, the rate of predation increases by ? (x, y) at a constant k. from the above equations, x (t) = ?(x) ? ?(x, y) and y (t) = ? ? (x, y) ? ?(y), there is both the constant (k) and unpredictable functions { ?(x), ?(y) and ? (x, y)}which are said to be dependent on several factors which are hard to be controlled. Discussion on the Applicability of Various Stability Models in Predator-Prey Relationship The Lotka-Volterra Predator-Prey Model Generally, the Lotka-Volterra Predator-Prey Model is represented by the equations x (t) = ?(x) ? ?(x, y) and y (t) = ? ? (x, y) ? ?(y). This equation can be reduced to classical Lotka-Volterra Predator-prey Model by simply introducing constants a, b, and c on the unpredictable functions [?(x), ?(y) and ? (x, y)] of the equation. (1.2) becomes x = ax ? bx y, ?y = ?bx y ? cy which is stable with predictable behaviours defined by x ? x_ ln x + B(y ? y_ ln y), however, the integral part of the equation implies that the prey has the capacity to grow infinitely in case the predator is not available which is biologically unrealistic as far as an ecosystem’s holding capacity is concerned. Introducing the maximum capacity that a given ecosystem can carry and the realistic growth capacity of the prey eliminates the infinite growth rate. This offers a finite equation: - ?(x) = ax(1 ? x/K), stability of the ecosystem in a predator prey model is obtained at the point where x = K while y = 0. The second model can be obtained by looking at the SIR and SIRS models in epidemiology. SIR model can also be obtained from the predator-prey model of equation [?(x, y) = ?x y +?y]. For the stable SIR model to be obtained, the variables x and y in the predator-prey equations have to be assumed to be fractions of the probability of the either the population to survive within the ecosystem in the presence of the other. The other assumption that has to be made is changing the unpredictable variable into predictable variables. Further equating w (0, y) ? 0 produces a stable SIRS model. The last model that has been modified to stability within an ecosystem with unpredictable factors influencing biological processes is the Model of Nutrient-Phytoplankton Interaction and a Chemostat Model. This model is largely concerned with estimation of the amount of nutrient and the organisms relying on the same for survival. It generally assumes x(t) to be the amount of nutrient and y(t) to be concentration of the phytoplankton depending on the nutrients to give an equation of the form X (t) = (a – bx) – ?xy and y (t) = k (?xy) - cy Properties of the Predator-Prey Stabilizing Model For the mathematical predator-prey model to effectively represent the biological processes taking place within an ecosystem, its variables have been taken to represent different aspects in the biological ecosystem. For instance, x (t) and y (t) are components of the equation which are usually assumed to represent the populations or densities or concentrations of the two species being studies within any given ecosystem. In most cases, this is either the predators or the consumers. In order to obtain a stable model that can predict the behaviour of a given species under investigation and having a symbiotic type of relationship, the solution for the mathematical equation must not produce a negative result. The first property of the model is its ability to predict the capacity for the prey to grow steady in the absence of the predator. Research shows that such capacity is based on the assumption that the prey is independent and does not need the predator for it to survive. Consequently, if all the predators were to be eliminated from any given ecosystem, then the prey will experience a steady growth because of the increased survival capacity. Such growth is mostly represented unrealistically by the Lokta-Volterra model as having unlimited growth capacity without considering the fact that there is a given carrying capacity that any given ecosystem can accommodate and therefore cannot stretch beyond it. Biologically, any ecosystem has to should remain at a given equilibrium, an aspect that Lokta-Volterra model cites to be a predator-free equilibrium detonated by Q0 = (x0, y0). On the other hand, the number of predators within any given ecosystem is regulated by the nature of the predator-free equilibrium (Q0). This is the component that is responsible for the regulation of the ecosystem to ensure that it remains stable at all time. For instance, if the predator-free equilibrium is asymptotically stable, then the predators will have low capacity for survival whenever they are introduced to any given ecosystem. On the other hand, a globally stable predator-free equilibrium would imply that the predators will only survive in the presence of the prey and eventually die when the prey become extinct. Basic Reproduction Rate (R0): This concept is concerned with the reproduction capacity of any predator. In most case when the R0 is greater 1, then it implies that such ecosystem has a higher probability to support the continuity of the predators. On the other hand, whenever R0 is less than 1, then it means that there would be limited chances for the predators to survive in such an ecosystem. From the above explanation, a basis reproduction number can be represented as:- R0 = {[? ??(x0, y0)] ? ?y} ? {[d?(y0) ? dy]} (2.2) Various States of Globally Asymptomatically Stable System Any ecosystem can display 3 varied globally asymptomatically stable states. This is shown as below, i. System with As positive equilibrium in which the equilibrium state is unique ii. System no positive equilibrium state has a predator-free equilibrium state denoted as Q0 = (x0, 0). iii. The basic reproduction number is perceived as the entrance point for all species being considered within an ecosystem where either positive or negative equilibrium occurs at R0 > 1 and R0 6 1 respectively. Stability of a Positive Equilibrium State There are certain aspects that indicate an equilibrium state in an ecosystem. This include equilibrium state being the only fixed point, there is non-positive change in auxiliary function as time changes, and finally in an event of zero result, there would always be invariant set except for the equilibrium state. Stability of the Predator-Free Equilibrium State: This state only exists in the presence of certain functions which include positive or negative equilibrium. It can be presented as: ?(x0) = ?(x0, y0) = B?(y0) ……………….. (4.1) The above equation (4.1) occurs in the presence of Q0 = (x0, y0), x0 > 0 and y0 = 0, and x0 and y0 satisfy. These parameters that control equilibrium-free state bust hold for stability to be attained. Ideally, the proposition in part (ii) that asserts that system with no positive equilibrium state has a predator-free equilibrium state denoted as Q0 = (x0, 0) can be justified through predator-free equilibrium state. From this understanding, various hypotheses can be formulated to justify a stable ecosystem. H6 hypothesis can be represented by ?(x) {1 ? limy>y0 [?(x0, y) ? ?(x, y)]} H7 hypothesis can be represented by ??(x0, y) ? ?(x, y) ?x > 0 for y > 0. H8 ?(x, y) ? ??{(x,0)??y y} for all x, y > 0. H9 ?(y) ? {d?(0) ? dy y} for all y > 0. H10 ?(0) = 0. In conclusion, there are specific mathematical models that have been found to be appropriate for studying the characteristics of predator-prey relationship within an ecosystem. This implies that there are many mathematical models for conducting such studies, but not all of them can be applied in the predator-prey study. One of the important aspects that biomathematics researchers consider before choosing a model of study is the ability of the model to be produce stable states that can effectively help in predicting the predator-prey relationship within an ecosystem. For a very long time, biomathematics researcher focused on density-dependent higher-linear mortality models instead of those that produce linear relationship. Such researchers based their choice on the assumption that such non-linear systems had higher chances of producing varied systems, a fact that has been disapproved by the basis reproduction rate RO. Often RO is believed to be an important parameter when it comes to regulating the properties of all systems found in the world. In fact, whenever the linear mortality of any system vanishes, basic reproduction rate has been found to tend towards infinite. This implies that if at all the basic reproduction rate is to be neglected in the choosing of mathematical models for studying predator-prey relationship then the researchers should be prepared to change the whole system to be studied. Moreover, results obtained from the mathematical tests are important in interpretation of biological processes. This is attributed to the fact that through mathematics, one is able to choose appropriate equation that will ensure that reliable results are obtained. A good example in situations where mathematical results have of great benefit to the biological implications is the approval of the systems with just small variation in shape have produced varied stabilities. Works Cited Korobeinikov, Andrei. Stability of Ecosystem: Global Properties of a General Predator-Prey Model. Mathematical Medicine and Biology. 26(2009): 309-321. Read More