Mathematical Model of Predator-prey Relationship With Human Disturbance
Chapter One
AIM OF THE STUDY
Based on the previous works done on investigations, contributions and modifications on predator-prey model, our aim and flair in this model is to find out the effect of human disturbance to the system and proffer solution to or solve the existing equations in two variables and analyze the obtained result. The model will tell us about the effect of human disturbance, periodic force, noise and diffusion. This will also show that the motion of individual species of the given population is random and isotropic that is no preferred direction. It will also analyze the state of the system in the presence of human disturbance and the predator’s functional response with the Holling Type-III response.
CHAPTER TWO
LITERATURE REVIEW FOR THE STUDY
The predator-prey model under review is the type that various scientists and mathematicians have investigated for some predominant factors and they decided to pen down their observations for further studies. Among them is the first simple predator-prey model called Lotka-Volterra Models in the year 1925-1926 which was carried out by Mahaffy [3]. They developed techniques for the qualitative analysis of nonlinear differential equations where the equation contained a single unknown variable. They also looked at a Predator-Prey model where two species are involved. They developed the mathematical models for two species that are intertwined in a predator-prey or host-parasite relationship. They analyzed to find the equilibria, discussed the numerical solutions of these systems of differential equations and the stability of the equilibria. They also used the simplified predator-prey interaction seen in Canadian northern forests where the population of the lynx and the snowshoe hare are intertwined in a life and death struggle.
The system of equations in predator-prey model is an example of a Kolmogorov model Freedman [4]. In 1920 Lotka extended, via Kolmogorov the model to “organic systems” using a plant species and a herbivorous animal species as an example and in 1925, he utilized the equations to analyze predator-prey interactions in his book on biomathematics (Volterra.V.1926); which brought about the popular predator-prey equation. Vito Volterra made a statistical analysis of fishes caught in the Adriatic and investigated the equations in 1926. Hongler and Filliger [5] presented an exactly soluble predator-prey model for two species interacting according to a non-linear Kolmogorov-type of equation. They obtained simple analytical expressions for the parametric equations of the cycles, the cycles in the phase space, the Hamiltonian and the corresponding integrating factor showing that the dynamics is globally conservative. They also formulated a general method to construct similar classes of population models. As a particular example, they derived the evolution equation describing the time dependent behavior of a two-player, two-strategy asymmetric game. Wiens [6] stated the Lotka-Volterra system of equations which is a more general framework that can model the dynamics of ecological systems with predator-prey interactions, competition, disease and mutualism.
Besides, Tyson, Sheena and Hodges [7] modeled mathematically the Snowshoe Hare (lepus americanus) and Canada lynx (lynx Canadensis) population cycles in the boreal forest focused largely on the interaction between a single specialist predator and its prey. Here, they considered the role of other hare predators in shaping the cycles using a predator-prey model for up to three separate specialist predators. They also considered the Canada lynx, Coyote (lanis latrans) and great horned owl (Bubo virginianus). Their model improved on past modeling efforts in two ways. That is to say that their model solutions represented the boreal hare and predator cycles with respect to the cycle period; maximum and minimum hare and predator densities for each predator. They also sheds light on the role of each specialist in regulation of the hare cycle, in particular, the dynamics of the predator appeared to be crucial for characterizing the low hare densities correctly. Sibenaller [8] investigated a simulation that illustrated how predator-prey interactions affect population sizes and how competitive interactions affect population sizes. They simulated the interactions between a predator population of fox and a prey population of rabbits in a meadow. After collecting the data, they graph the data and then analyzed the graph to predict the populations for several more generations. They also examined the co-evolutionary interaction between predator and prey (how predators react to selective pressure by increasing their efficiency and how prey becomes more skillful at evading their predators).
Sun et al [9] investigated the effects of migration in a predator-prey system with self diffusion. They also determined the features in the dynamics that are specifically caused by migration terms and they chose nonlinearity in the local kinetics taking the form of a non-dimensional Holling-Tanner system (Holling type-II function). In their paper, they gave a Holling-Tanner model with both migration and diffusion in which the parameters are rational biological that determined the dynamics of the system. They analyzed the dynamics of the model and derived the dispersal relation with respect to the model parameters. They performed simulations by illustrating the emergence of travelling pattern and calculated the wavelength and pattern speed. Properties of pattern solutions were revealed. Sun et al [10], carried out an investigation on a spatial version of the predator-prey model with Holling III functional response which includes some important factors such as external periodic forces, noise and diffusion processes. The influence of periodic force and noise which was less understood if considered in space was addressed by the means of mathematical modeling and extensive computer simulations. They also focused attention on a predator-prey system with Holling III functional response that is to ascertain the predator’s rate of feeding upon the prey. Not only this, these proponent researchers identified the spatial component of ecological interactions as an important factor in how ecological communities are shaped. Thus, in their paper, spatial pattern of a spatial predator-prey model was combined with migration and diffusion was investigated. They presented the emergence of spatial pattern by both theoretical analysis and numerical simulations. Their study revealed that migration had great influence on the pattern formation. More specifically, they stated that travelling pattern could be induced by it. Furthermore, wavelength of the travelling pattern with respect to the diffusion coefficient of the predator was calculated. Their results extended the finding of pattern formation in the ecosystems and may well explain the field observed in some areas.
CHAPTER THREE
THE MODEL
In this predator-prey model with human disturbance, the forcing functions are:
Human Disturbance
Noise.
External Periodic Force
However, we will be looking at one of the most important components of the predator-prey relationship that is the predator’s rate of feeding upon the prey which is the predator’s functional response. These functional responses can be either prey-dependent or predator-dependent. Thus, the functional response equations that are strictly prey-dependent are the Holling group. From the predator’s and prey’s equation below:
(a)
(b)
Using equation (a) the constants,a1= u ( natural growth rate of the rat denoted as, ur).a2 =
(denotes the carrying capacity of the Rat). Because the human predation is negative on both species and the prey is mainly affected we summed it up to give us ,a3 =b3 = Eq ( the predating coefficient and total effort applied on predating on the population of the rat, r plus the Holling term. This implies that : The rate of change of Rats population with respect to change in time, is equal to the natural growth rate of the Rat, ur minus the product of the natural growth of the Rat ,ur and the carrying capacity of the Rat (prey) in the absence of the Cat (predator) and Human Predation, and a reduction in the functional response of the predator on the prey,and a reduction in the product of the predating catch ability coefficient of the Rat species, q and the total effort applied on predating on the Rat population, Er. By these definitions, the model for Rat population becomes:
CHAPTER FOUR
DISCUSSION OF THE RESULTS
In this model, we showcased a predator-prey interaction with the predator’s functional response as the Holling Type-III functional response which includes some vital factors like human disturbance, external periodic forces or oscillation, diffusion and noise. We used scilab in the simulations as shown in figures 1 to 11 below. We determined that human disturbance and noise can lead the whole system to show the oscillatory waves pattern which is seen in figure 1. Figure 1 is the population of the predator (Cat) and prey (Rat) against time with human disturbance and noise. Different initial values lead to different trajectories of typical time series of predator C(t) (dotted line) and of prey R(t) (solid line). As a consequence, solutions show oscillations with a frequency that is given by (as shown in Fig.1). We can say that the common sense explanation for the origin of the oscillations is delayed predator-prey interactions and the emergence of human disturbance.
However, as the intensity of noise and human disturbance are increased, the fluctuation is apparent. It is clear that the frequency of the waves and resounding pattern arise when human disturbance, noise and external periodic forces are present in the system (equations 5 and 6). We also found that a regular frequency pattern is a general characteristic of predator-prey model. Both species are at equilibrium at some time and population density as time and population keep increasing (see figure:1). Moreover, figure:1 is independently showing the population of each species with time; but in contrast, both portray that as the predator eats up the prey and its population density reduces at some time interval, the predator’s population starts increasing at some time interval also and when there is no much prey for the predator to eat its population density starts dwindling while the prey’s population density starts escalating to its maximum height and this happening is in a continuous process. This same figure also exposed that both species are in equilibrium at the point of intersection at some population density and time interval. The prey population against time shows a higher, stronger and clearer sinusoidal wave pattern than the predator’s population against time. Human disturbance has an immense effect or influence in the model in that it supports the modulating nature of the species especially on the prey population. Its presence makes the prey’s solutions in the result for discussion to be in a declining pattern; see figure 1. The said factor decreases the population density of the prey even in the absence of the predator or when the predator has not been introduced into the competition or system. It is the presence of the human disturbance that makes figure 1 to show a falling and rising nature. Figure 1 also expounds that the incidence of human disturbance leads the prey to swift annihilation at the commencement of the competition. For the prey to rise up again for the contest, it is the resonance of the predators’ noise that quarantines the prey in its hide-out which enables it (prey) increase in number through giving birth to the young ones.
Figure 2 end result is the population of the predator against prey population with human disturbance and noise. The existence of human disturbance gives rise to exponential rise in the e population densities of the prey and predator. In terms of the prey, its harm is directly done by killing them off from the contest while in turns causes starvation to the predator as its food is being destroyed and that leads to the death of the predator from the competition. From figure 2, it depicts that a reduction in the population size of one species leads to a reduction in the population size of the other and an increase in the population mass of one species leads to an increase in the population mass of the other species at different time interval and population size as the time and population increases; and vice versa. The figure is also telling us that when predator’s density is 1, prey is 2; taking the ratio gives 1:2 and when the density increased to 2:10 the ratio becomes 1:5.
CHAPTER FIVE
SUMMARY
The predator-prey relationship with human disturbance and other factors such as noise, diffusion and external periodic force is considered in the model. The functional response of Holling III is also involved in the study. This predator-prey model involves two species giving us two variables (the predator and prey). The oscillatory wave in two-dimensional space is shown by the species with time which is obvious when human disturbance and noise are involved. In this model, the coefficient of diffusion is zero at the point predator is predating on the prey. Also, the effect of the said factor (human disturbance) leads the prey to quick annihilation from the system of interaction at the beginning of the competition and later comes up in its population in an asymptotic and exponential increase respectively. The study when modeled with noise and periodic force showcased a sinusoidal and an exponential increase in the figures below; and without noise and periodic force depicted an asymptotical increase in the shape of the graph figures below. These results may help us to understand the effects springing up from the true defenselessness to random fluctuations in the real ecosystems. We declared that the human disturbance increases the functional response and the entire processes of motion (diffusion) which showed us that the predator has only one type of food source. Both the prey and predator will survive the contest. The study has showcased the rate of the predator’s functional response with time, t. We analyzed and discussed the equilibria, stability of the model and solutions of these systems of differential equations. We also used the figures to illustrate the predator-prey interaction in terms of their population which exists in an ecosystem, predator-prey life in an ecological system, a predator predating on its prey and the intensity of human disturbance in the same ecosystem. We performed simulations by illustrating the rate of the predator’s feeding on the prey with time using the Holling-Type III functional response showing the searching time, handling time and total time of the predator in predating on its prey. We used scilab in the simulations as shown in figures 1 to 15.
CONCLUSION
Conclusively, we investigated that human disturbance is a form of noise induced into the predator-prey system during competition. In this model, when human disturbance and the noise term are considered in the system, the oscillatory waves (oscillations) of the species involved in the system of interactions become clearer and these increase the intensity of noise experienced in the whole system. Nevertheless, the positive solutions of the two variables (predator and prey) equation means that both species will survive the contest but only starvation will lead the predator go to extinction. Figure 1 to 6 also expounded that the incidence of human disturbance leads the prey to swift annihilation at the commencement of the competition and later grows up in asymptotical and exponential pattern respectively. For the prey to rise up again for the contest, it is the presence of the predators’ noise that quarantines the prey in its hide-out which enables it (prey) increase in number through giving birth to the young ones. Therefore, the effect of human disturbance in this study shows the oscillatory wave or periodic force wave front of the predator and its prey to be stronger in figure 1, depicts exponential increase in figure 2&3 and asymptotical increase in figure 5&6.
RECOMMENDATION
In view of the fact, as shown in this model, that the predator is potentially a more useful biological organism control if other things being equal, it should be established faster. This model should be used by the ecologists to eliminate the unwanted organisms called prey in an ecosystem. Since the predators are helpful to the ecologists in determining the kind of organisms that should exist in an ecological system. In ecology, they act as scavengers. They are not there to serve humans, but their position on the trophic ladder is important. With regards to laboratory use (medically), their metabolic systems are virtually identical to humans. Thus, when a drug is tested it is frequently tested on rats.
Generally, predator-prey model could be applied in divers areas of real life situations such as in economics, it is used to link the population of various industries by introducing trophic functions between various sectors and ignoring smaller sectors by considering the interactions of only two industrial sectors and check inflation. Agriculturally, it can be applied in fishery management. In physics and geography, it can be applied using aerosol–cloud–precipitation system. Commercially, it can also be applied on the land transport to scare robbers attack. That is to say that when more vehicles are travelling together they will outweigh the fear of robbery assault.
AREAS OF FURTHER RESEARCH
The study can be carried out with larger sample population of predator and prey, taking the age of both the predator and prey into consideration. Considering two predators that can kill and eat, and also eat dead preys.
REFERENCES
- Wallace, R. A. (1979): The Ecological and Evolution of Animal Behaviour. 2nd ed. Scott, Foresman, Glenview. New York.
- Wallace, R. A. (1991): Biology the Science of Life. 3rd ed. Harpercollins publishers Inc. New York.
- Mahaffy, J. M. (2000): Lotka-Volterra Models.Williams and Wilkins Co. San Diego State..
- Freedman, H. I. (1980): Deterministic Mathematical Models in Population Ecology. Marcel Dekker. New York.
- Hongler, M. O. and Filliger, R. (2005): An Exactly Soluble Kolmogorov Model for two Interacting Species. Springer, New York.
- Wiens, E. G. (2003):Lotka-Volterra Equation. file://F:/Lotka-Volterra _equation predator prey.htm
- Tyson, R, Haines, S and Hodges, K. E. (2009): Modelling the Canada Lynx and Snowshoe Hare Population Cycle: the role of specialist Predators. Springer Science + Business Media B.V.
