Mathematical model of predator-prey relationship with human disturbance

Project Abstract

<p> The predator-prey model with human disturbance is considered in the model and other<br>factors such as noise, diffusion and external periodic force. The functional response of Holling<br>III is also involved in the study. This predator-prey model involves two species giving us two<br>variables (the predator and prey). The oscillatory wave in two-dimensional space is shown by<br>the species with time which is obvious when human disturbance and noise are involved. In this<br>model, the coefficient of diffusion is zero at the point predator is predating on the prey. Also, the<br>effect of the said factor (human disturbance) leads the prey to quick annihilation from the system<br>of interaction at the beginning of the competition and later comes up in its population in an<br>asymptotic and exponential increase respectively. The study when modeled with noise and<br>periodic force showcased a sinusoidal and an exponential increase in the figures below; and<br>without noise and periodic force depicted an asymptotical increase in the shape of the graph<br>figures below. These results may help us to understand the effects springing up from the true<br>defenselessness to random fluctuations in the real ecosystems. We declared that the human<br>disturbance increases the functional response and the entire processes of motion (diffusion)<br>which showed us that the predator has only one type of food source. Both the prey and predator<br>will survive the contest. The study has showcased the rate of the predator’s functional response<br>with time, t. We analyzed and discussed the equilibria, stability of the model and solutions of<br>these systems of differential equations. We also used the figures to illustrate the predator-prey<br>interaction in terms of their population which exists in an ecosystem, predator-prey life in an<br>ecological system, a predator predating on its prey and the intensity of human disturbance in the<br>same ecosystem. We performed simulations by illustrating the rate of the predator’s feeding on<br>the prey with time using the Holling-Type III functional response showing the searching time,<br>handling time and total time of the predator in predating on its prey. We used scilab in the<br>simulations as shown in figures 1 to 15.<br>Key Words predator-prey model, human disturbance, external periodic force and noise. <br></p>

Project Overview

<p> INTRODUCTION<br>Predation is the process of removing individuals from a lower trophic level as to<br>prevent monopoly competitive success among the prey. Predation thus allows increased<br>diversity through what is called ‘‘cropping principle’’. This effect is demonstrated by removing<br>top predators which results in drastic reductions in prey diversity as successful competitors freed<br>from predation preempt resources. Predation can have a major effect on the size of a population<br>as applied to population that when the death rate exceeds the birth rate in a population, the size<br>of the population usually decreases. If predators are very effective at hunting their prey, the<br>result is often a decrease in the size of the prey population. But a decrease in the prey population<br>in turn affects the predator population. Wolves and Lions preying on ungulates, and Cats preying<br>on Rats have their take limited by the effective defenses of the prey animals such that their<br>predation cannot interrupt rapid population growth of the prey when food and population<br>dynamics produce exponential increase, but relatively high predator densities accentuate<br>population crashes that follow. Predation can be a powerful determinant of community structure.<br>It has a dynamic influence on the numbers and quality of both predator and prey as it acts as an<br>important agent of natural selection on both groups.<br>However, diversity in ecology is the measure of the number of species coexisting in a<br>community. An ecosystem is a system of plants, animals and other organisms interacting within<br>themselves and non-living components of their environment; e.g. a lake or forest. There are<br>‘‘natural’’ and ‘‘managed’’ (that is farms or market gardens) ecosystems. Today, few<br>ecosystems remain untouched by human activities. Managed ecosystems are essential to our<br>survival by reducing competition through removal of non-useful species (that is weeds). People<br>are able to intensify food and other natural materials production. These processes more often<br>reduce species diversity but there are instances where human management of ecosystems<br>actually increases species diversity. No simple relationship exists between the diversity of an<br>ecosystem and ecological processes. An ecological system is an open system in which the<br>interaction between the component parts is non-linear and the interaction with the environment is<br>noisy. The model will explain the interaction between the species and their natural environment<br>which is the ecological system.<br>2<br>Nevertheless, the predator–prey model is the building blocks of the bio and ecosystems as<br>biomasses are grown out of their resource masses. The predator–prey model is a type of<br>mathematical model that involves at least two species (the predator-cat and prey-rat). In the<br>course of the species existence, the species involve compete, develop or evolve and scatter or<br>disperse for the purpose of searching for resources to sustain their living. Based on their specific<br>settings of applications the predator–prey can take the forms of parasite-host, tumor cells (virus)–<br>immune system, resource–consumer, plant–herbivore etc. The predator–prey embark on the<br>business of one specie’s loss is another specie’s gain; interactions may have applications outside<br>the ecosystems.<br>In the biological point of view, the first rush of ecological theory saw predators as potential<br>controlling agents for populations. Indeed, predators can utterly transform population histories;<br>but the more interesting effects are probably on diversity and structure as predator winnowing of<br>populations alters patterns of competition. It is a truism of history that much of the food of<br>wolves and big cats consists of the old and the sick.<br>1.2 AIM OF THE STUDY<br>Based on the previous works done on investigations, contributions and modifications on<br>predator-prey model, our aim and flair in this model is to find out the effect of human<br>disturbance to the system and proffer solution to or solve the existing equations in two variables<br>and analyze the obtained result. The model will tell us about the effect of human disturbance,<br>periodic force, noise and diffusion. This will also show that the motion of individual species of<br>the given population is random and isotropic that is no preferred direction. It will also analyze<br>the state of the system in the presence of human disturbance and the predator’s functional<br>response with the Holling Type-III response.<br>1.3 DEFINITION OF TERMS IN THE STUDY<br>The following terms will be defined in this section:<br>(i.) The predator and Prey<br>(ii.) Human disturbance<br>(iii.) Oscillation or periodic force<br>3<br>(iv.) Noise<br>(v.) Diffusion<br>THE PREDATOR AND PREY<br>A predator is an organism that uses other live organisms as an energy source and in doing<br>so, removes the prey individuals from the population. This definition allows the concept of<br>predation to be extended to include herbivore as well as carnivore. The working ecologists now<br>talk of predation when describing sheep hunting grass, cats hunting rats or squirrels searching for<br>nuts. When predators kill, they remove contestants in an ecological game. This changes the rules<br>for all the other players. If a competitor is taken out, those that are left benefit. Just like when a<br>seed dispersal agent is removed, a plant is not transported. If an enemy is killed, an old victim<br>flourishes. Predators are in a sense arbiter of community structure and local diversity.<br>In this study, a predator is an animal that hunts, kills and eats other animals for example<br>Lion, Cat, Wolves and other predators. The predator is a carnivorous animal and the prey is a<br>herbivorous animal. An example of simplified predator-prey interaction in our environment is<br>seen in a house where Rats and Cats are living. The population of the Cat and Rat are intertwined<br>in a life and death struggle or fight .It had been predicted by the ecologists that in a sample<br>predator–prey system, that a rise in prey population goes with a move slowly (with a lag) by a<br>rise in the predator population. When the prey population falls, the predator population falls and<br>this allows the prey population to recover and complete one cycle of this interaction.<br>Predators influence the numbers of prey by removing individuals from the prey<br>population, yet they do not kill off the prey population. This is because under undisturbed<br>conditions, prey population rise steadily thus providing more food for predators. Then the<br>predator population begin to rise, their numbers do not rise immediately since it takes time for<br>the energy from food to be converted into successful reproductive efforts. Because of this time<br>lag, the prey may be well on the road to recovery before the predator population begins to rise.<br>When the predator population finally rises, there is increasing pressure on the prey. Then as the<br>prey begins to be killed off, the predators find themselves with less food and so their own<br>population soon falls off due to starvation or simply a failure to reproduce; this helps the prey to<br>4<br>recover. The predator-prey relationships are not often straight forward. Below is the picture of<br>cat pursuing rat.<br>FIGURE 1<br>HUMAN DISTURBANCE<br>The effect of human disturbance on the number of species found in the system is<br>recognized in the intermediate disturbance hypothesis. According to the hypothesis, areas with<br>intermediate levels of disturbance have more species than the areas of lower or higher levels of<br>Figure 1:<br>5<br>disturbance. At lower levels, competition is intense and the resulting exclusion yields only a few<br>surviving species. At higher levels, the disturbance itself wipes out all but a few stress-tolerant<br>species. At intermediate levels, not strong enough to kill most species but still strong enough to<br>reduce the competitive impact of dominant species, the number of species is the highest because<br>competitively inferior and superior species as well as stress-intolerant and stress- tolerant species<br>survive. Below is the intermediate disturbance hypothesis showing the number species plotted<br>against the frequency or intensity of disturbance. The diagram is illustrating that greatest<br>number of species found at intermediate frequencies or intensities of disturbance.<br>Figure 2: An Intermediate Disturbance Hypothesis<br>OSCILLATION OR PERIODIC FORCE<br>From the beginning, it will be seen that interaction of the two results in oscillations of<br>constant amplitude which is the time taken for system to ‘‘go round’’ one of the cycles is<br>Number of species<br>Frequency or intensity of disturbance<br>6<br>determined by the prey reproductive rate and the predator death rate. The different cycles<br>representing different amplitudes of oscillation result from the use of different values of an<br>integration constant which depends on the relationship between the rates of increase of the two<br>species. On the common sense grounds, the system of two species would continue to oscillate<br>with constant amplitude. In the more general treatment, what happens in nature is that while the<br>period of oscillation remains constant, the ratio of reproductive rates does not and thus the<br>amplitude of oscillation tends to change progressively so that the system either unwinds, the<br>oscillations becoming greater and greater until one species reaches zero and the system collapses<br>or alternatively, the oscillations tend to die down and the system comes to rest at the singular<br>point in the centre. In the first case, the collapse of the system, if the prey is the species to die out<br>the predator will rapidly follow suit. If the predator is the first to reach zero, the prey population<br>will increase until controlled at a new level by a new density-dependent factor such as food<br>shortage. The implication of this result is that, as soon as the population of one species reaches<br>zero, the whole system collapses. This theory is attributed to the small population involved and<br>the greater liability of the systems becomes extinct as a result. Oscillation will occur depending<br>only on the coefficients of increase of predator and prey and on the initial relative numbers.<br>The graph that comes from the records of pelts kept by the Hudson’s Bay Company in<br>Canada that is figure (3) will be used to show the oscillating rates of the predator-prey<br>relationship, Wallace [1, 2]. The peaks and crashes in the cat population are definitely dependent<br>on the rat population. The rat population is shown to follow a comparable pattern even in the<br>areas where there are no cats. The rats are responding to cycles in their own part ‘prey’, which<br>themselves seem to reflect climatic variations and changes in insect pest populations. The<br>interrelationships of predator-prey populations are clearly not as simple as it might first be seen.<br>See the diagram below:<br>7<br>Figure 3: Above is a graph of some analyses of fur data from the records of the Hudson Bay<br>Company in Canada according to Wallace [1,2].<br>The graph above shows the number of rats and cats living in ‘‘NDAAH PACKING<br>SHORE IN OMUANWA’’ from May 2011 to February 2012. In the graph of Cats and Rats,<br>from May to September, the number of prey (rat) increased. The Cats now had enough to eat, so<br>more of them survived. The growing number of Cats killed more and more Rats. The Rats<br>population decrease. By October, the lack of Rats had greatly affected the Cats. Some Cats<br>starved and others could not raise as many young. Soon the Rats population began to climb<br>again. This cycle for the two species has continued. This shows that the populations of Cats and<br>Rats are related. The Cats population depends on the size of the Rats population and vice versa<br>8<br>or predator-prey interactions. The oscillation of numbers of the predatory Cat and its prey (Rat)<br>is almost classic in its characteristics. Prey numbers increase first, followed at once by predator<br>numbers. Then as the predator increase continues, the prey species diminishes in a population<br>crash. The predator’s outcome is relatively similar as its number fall rapidly.<br>NOISE<br>Noise is a sound especially when it is unwanted, unpleasant or loud. In this case the<br>noise we need for an effective system is wanted and pleasant. The factor noise seen in this<br>system is exhibited by the predator and prey; when the battle of survival commences. Here, the<br>noise made by the prey is not sustained as to compare with that of the predator. On the part of the<br>prey, it sounds when trying to defend itself and it is caught by its predator. The prey’s noise<br>attracts the predator to itself while the predator’s noise scares them away to their hide-out. Once<br>it is in the domain of predator, killed the noise is terminated because a dead Rat does not make a<br>noise. However, the noise that comes from the predator is from the time of struggling to get hold<br>on the prey to when it is eating the prey; even after that the noise is still sustained and that is the<br>kind of noise that will be evident in our model equation. In this hint, noise in a system increases<br>the dynamics. In this model, the noise in predator’s equation will offset the noise in prey’s<br>equation.<br>DIFFUSION<br>Diffusion in this scenario is the process of movement of the predator and prey being<br>spread out and not directed in one place because of the chaos nature of the system. Diffusion is<br>regarded to be the motion of the species in the system. The diagram below depicts that there is<br>no motion at the time zero, which is the point of predation on the prey (Rat). The effect of<br>diffusion in the system is to initiate a travelling wave front which resulted in a smooth travelling<br>wave front solution for the reaction-diffusion equation.<br>9<br>Figure 4<br>10<br>ASSUMPTIONS<br>In this predator – prey model, we have the basic assumptions for the dynamics of the<br>populations of a predator and its prey species.<br>Let R(t) be the population density of the Rats (prey) and C(t) be the population density of the<br>Cats (predator) at time t. Thus, we want a mathematical model based on the growth rate for the<br>populations. The models C(r) of functional response are assumed to be continuously<br>differentiable on [0, ∞] and satisfy C(0) = 0, C1(r) ˃ 0 and<br>limı →∞ıııı= ı &lt; ∞<br>Such models include:<br>(i) ( )<br>r<br>p<br>k<br>r<br>r p<br>kr<br>p r<br>C r kr<br>+<br>=<br>÷ø<br>ö<br>çè<br>æ +<br>=<br>+<br>=<br>1 1<br>(ii) ( ) 2<br>2<br>p r r<br>C r kr<br>+ +<br>=<br>e<br>(iii) ( ) 2<br>2<br>p r<br>C r kr<br>+<br>=<br>where k, p and ı are positive constants. k denotes the growth rate of the species and p is the<br>saturation constant. In population dynamics, a functional response of predator to prey density<br>refers to the change in the density of prey attached per unit time per predator as the prey density<br>changes. C(r) denotes the predator response function. Based on the above functional response<br>assumed, (i) is called the Michaelis-Menten or Holling type-II, (ii) is called the sigmoidal<br>response function and (iii) is called the Holling type-III function and it satisfies the assumptions<br>made and will be used for the model. Type III functional response is the type in which the attack<br>rate accelerates at first and then decelerates towards satiation. Type III functional responses are<br>typical of generalists natural enemies which readily switch from one food species to another<br>and/or which concentrate their feeding in areas where certain resources are most abundant<br>11<br>This rate of change in a population with time is equal to the net increase (births) into the<br>population minus the net decrease (deaths) of the population.<br>Applying these assumptions to the population of Rats, the Rats population density<br>becomes the rate of change of the Rats population with respect to change in time (<br>dt<br>dR ). The<br>primary growth in the Rat population grows in proportion to its own population or size is a1R(t).<br>Hence, we will assume that the primary loss of Rats is due to predation by the Cats and Human<br>Beings (Human Disturbance). However, predation is often modeled by assuming random contact<br>between the species in proportion to their populations with a fixed percentage of those contacts<br>resulting in death of the prey species. Mathematically, this is given by a negative term − a2<br>R(t)C(t) and − a3R(t). Here, a3 is the total number of prey, n minus the number predated by<br>Human Being, m i.e. (n–m). If the Cats population is low such that starvation due to<br>overcrowding dominates the death rate and no human disturbance, then alternative death terms<br>would be more appropriate. Combining these terms, we have the growth model for the Rat<br>population:<br>a R(t) a R(t)C(t) (n m)R(t)<br>dt<br>dR = – – – 1 2<br>( ) 1 2 a R t a<br>dt<br>dR = – ( ) R t C(t) a R(t) 3 – (This is the prey’s equation).<br>For the predator, we will consider the population dynamics of the Cat as the rate of change<br>of the Cats population with respect to change in time (<br>dt<br>dC ). The primary growth for the Cat<br>population depends on sufficient food for raising Cats, which means an adequate source of<br>nutrients from predator or preys. Thus, the growth of Cat population is similar to the death rate<br>for the Rat population with a different constant of proportionality.<br>Mathematically, the growth of the Cat population can be expressed as b2 R(t)C(t). The human<br>predation is negative on the Cat because it reduces the quantity of food meant for the predator<br>i.e. (n – m)C(t) = b3C(t), where b3 represents the total number of Rats, n minus the number<br>predated by human beings, m. The loss of Cat is presumed to be a type of reverse growth. That<br>12<br>is, in the absence of Rats, the Cat population declines in population to their own population<br>which is expressed by the negative modeling term as − b1 C(t).<br>If the Cat diet used other animals or crowding factors from other predators and Cats were taken<br>into consideration or account, then these terms would have to be significantly modified. The<br>growth model for the Cat population gives:<br>b C(t) b R(t)C(t) (n m)C(t)<br>dt<br>dC = – + – – 1 2<br>b C(t) b R(t)C<br>dt<br>dC<br>1 2 – + (t) b C(t) 3 – This is the predator’s equation.<br>The model ignores the role of climate variation and the interactions of other species. Other<br>significant factors ignored are the ages of the animals and the spatial distribution. The two<br>differential equations above are intertwined into a system of differential equations with each<br>growth model depending on the unknown variable (population) of the other.<br>13 <br></p>