A NUMERICAL STUDY ON PREDATOR PREY MODEL

Transcription

1 International Conference Mathematical and Computational Biology 2011 International Journal of Modern Physics: Conference Series Vol. 9 (2012) World Scientific Publishing Company DOI: /S A NUMERICAL STUDY ON PREDATOR PREY MODEL MOHAMED FARIS LAHAM Institute for Mathematical Research, Universiti Putra Malaysia mohdfaris@putra.upm.edu.my ISTHRINAYAGY KRISHNARAJAH Institute for Mathematical Research, Universiti Putra Malaysia isthri@math.upm.edu.my ABDUL KADIR JUMAAT Faculty of Computer & Mathematical Sciences, Universiti Teknologi MARA Shah Alam abdulkadir@tmsk.uitm.edu.my Stochastic spatial models are becoming a popular tool for understand the ecological and evolution of ecosystem problems. We consider the predator prey interactions in term of stochastic representation of this Lotka-Volterra model and explore the use of stochastic processes to extinction behavior of the interacting populations. Here, we present simulation of stochastic processes of continuous time Lotka-Volterra model. Euler method has been used to solve the predator prey system. The trajectory spiral graph has been plotted based on obtained solution to show the population cycle of predator as a function of time. Keywords: Stochastic; predator prey; Euler method; Lotka-Volterra model. 1. Introduction One of the dominant themes in ecology is the dynamic relationship between predator and their prey. Predator-prey interactions have been studied since as early as 1925 when mathematical model were proposed independently by Lotka and Volterra leading to the Lotka-Volterra model 1. The Lotka-Volterra model arises in mathematical biology and models the growth of animal species as discussed in many mathematical biology books for example Ref. 4, 5 and 6. Interest often lies on studying the dynamics of the equilibrium solution of the process. 347

2 348 M. F. Laham, I. Krishnarajah and A. K. Jumaat Population cycles of prey and their predators are described by frequent increase and decrease in the densities of a predator and its prey. In general, higher prey population size than the predator will give a realistic ecological dynamics; whereas, higher predator population size will give a rapid extinction of predator population. One of the famous examples of predator-prey interactions was illustrated by Canadian lynx as predator and snowshoe hare as prey 3. The dynamics of the hare cycle helps the ecologist to understand the importance of predation and food supplies in regulating that cycle. Cycles are useful in a given time period because the changes can be predictable with some confidence and more changes may occur than in an irregularly fluctuating population. Any fluctuating population can be understood by knowing first in detail the mechanism of changes in births, deaths and movements which are the proximate causes of the changes in numbers. The predator s fluctuating population follows those of the prey population through time. However, stochastic models are becoming an important mathematical tool for understanding ecological complexity and play a fundamental role in predator prey population processes 2. It helps to explain observable population phenomena. A stochastic process is a natural model for describing the evolution of real-life processes, objects and systems in space and time 4. It presents contributions on the mathematical approaches from structural, analytical and algorithmic to experimental methodologies. It also offers an interdisciplinary simulation on the uses of probability theory and discusses real-world applications of stochastic models to various areas such as biology, queuing theory, inventories and dams, storage, computer science, telecommunication modeling, reliability, and operations research. Stochasticity is the extensions to differential equation models and there are definitely many lessons for artificial life researchers in these equations. Most likely, both techniques will have an important role to play in further study of population dynamics. In this paper, simulation of stochastic processes of continuous time Lotka-Volterra model is presented. Euler method has been used to solve the predator prey system. The trajectory spiral graph has been plotted based on obtained solution to show the population cycle of predator as a function of time. 2. The Deterministic Model Deterministic mathematical models have frequently been used as tools in the study of species interaction in an ecological situation. In the classical representation of predator prey interactions as formulated by Lotka-Volterra model, the prey population grows exponentially in the absence of predators. This is based on the assumption that there is always sufficient food for the prey and there is no other threat to its growth.

3 A Numerical Study on Predator Prey Model 349 The model involves two mathematical equations since we are considering two species, one describes how the prey densities changes and the second which describes how the predator densities changes. Consider two species where H ( t) denotes the population of prey and P( t) denotes the population of predator at time t. Suppose that in the absence of predators, prey increase at rate α (which, for stochastic simulations, is calculated as the birth rate b minus the natural prey death rate), whilst in the absence of prey, predator die at rate γ. A particular case of the Lotka-Volterra differential system are given as: dh / dt = α H β HP dp / dt = λhp γ P Here within-species competition has been ignored. The constant β measures the death rate of prey due to being eaten by predators; the prey population will be depleted faster when the number of predator population is greater. The constant λ measures the skill of the predator in catching prey; the availability of predator food resource will be greater when the number of prey is greater. The solution for the deterministic Lotka-Volterra model represents sustained oscillations for positive constants,,, α β λ γ along the closed curves γ log H λh + α log P β P = const excepts if initially the system does start in the equilibrium point = / and H = /. α β 3. A Stochastic Lotka Volterra Model P γ β The stochastic Lotka-Volterra takes into version the discreteness of the population and their random fluctuations. In the stochastic model, extinction of species is possible and depending on the initial conditions. The dynamics of the system can deviate drastically from the deterministic model. The stochastic model for predator prey system is formulated as follows: P (Prey death N N 1) = bpnδ t P (Prey birth N N + 1) = ( a rn ) N t P (Predator birth P P 1 P (Predator death P P 1 + ) = cpnδ t ) = dpδ t δ

4 350 M. F. Laham, I. Krishnarajah and A. K. Jumaat Where the constants N and P are the time dependent prey and predator population sizes respectively. a and r is the prey birth rate parameter and d is the predator death rate parameter. Thus in the absence of prey, the predator population decreases exponentially with rate c. In the presence of both species, interactions are modeled as proportional to the product of the sizes of prey population and predator population. The interaction results in prey being eaten by predator with death rate of the prey bp( t) N( t ). The survival of the predator population depends solely on the prey. The greater the number of prey shows the greater the availability of food for the predators. Thus the birth rate for predator is cp( t) N ( t) and the ratio c / a measures the efficiency of predation in terms of birth of predators. In the absence of predators, the prey population exhibits logistic growth with rate a and 2 carrying capacity a / r. Without the term rn, the prey will grow exponentially at rate a in the absence of predators. However, this term ensures that the prey population never exceeds the carrying capacity. 4. Numerical Simulations To demonstrate the numerical solution of the stochastic predator prey model, we set up three initial conditions; P (0) = 12, H (0) = 1000 ; P (0) = 15, H (0) = 1000 and P (0) = 18, H (0) = Figure 1 and Figure 2 show the solution using Euler method for t up to t = 1. Mainly we simulate the trajectories graphs of the predator densities versus prey densities. The system has been solved numerically using Euler method and was determined using Matlab software. Based on the graph, when the number of predator initial population is increase, then the shape of cycle graph becomes smaller.

5 A Numerical Study on Predator Prey Model 351 Fig. 1: Numerically solve a predator prey model

6 352 M. F. Laham, I. Krishnarajah and A. K. Jumaat Fig. 2: Stability for Euler s method in a simple predator prey model without noise

7 A Numerical Study on Predator Prey Model Discussion and Conclusion In Conclusion, this Lotka-Volterra Predator-Prey Model is a fundamental model of the complex ecology of this world. It assumes just one prey for the predator, and vice versa. It also assumes no outside influences like disease, changing conditions, pollution, and so on. However, the model can be expanded to include other variables, and we have stochastic Lotka-Volterra model, which models two competing species and the resources they need for survival. We can enhance the equations by adding more variables to get a better picture of the ecology. However with more variables, the equations becomes more complex and would require more mathematical analysis and numerical simulations. This model is an excellent tool to teach the principles involved in ecology, and to show some rather counter-initiative results. References 1. Alan, A, Berryman. Ecology. 73, 5 (1992). 2. Keeling M.J., Wilson H.B., Pacala S.W. Naturalists. 159, 1 (2002). 3. Krebs J.C., Boonstra R., Boutin S., and Sinclair A.R.A. Bioscience. 51, 1 (2001). 4. Renshaw E. Modelling Biological Populations in Space and Time. (Cambridge, 1991). 5. Murray J.D. Mathematical Biology I, An Introduction. (Springer-Verlag, New York 1993a). 6. Murray J.D. Mathematical Biology II, Spatial Models and Biomedical Applications. (Springer- Verlag, 1993b)