In this experiment, a simple Looks-Voltaire type model was chosen to demonstrate these behaviors. The concept of state trajectory, limit cycles and stability will be introduced. Introduction: There is a atrophic level controls the dynamics of another atrophic level is central to ecology. Energy flows between atrophic levels. The input of energy to a higher atrophic level population from a lower atrophic level population can be controlled by the amount of energy available in the lower level, or by the amount of energy the higher atrophic level is able to consume.

Or we take into consideration such factors as the “natural” growth rate and the “carrying capacity” of the environment in the dynamics of a single population. Mathematical ecology requires the study of populations that interact, thereby affecting each other’s growth rates. In this module we study a very special case of such an interaction, in which there are exactly two species, one of which the predators eats the other the prey. In this experiment, we will assume unrealistic case in these predator-prey situations. 1. The predator species (fox) is totally dependent on a single prey pieces(rabbit) as its only food supply, 2.

There is no threat to the prey other than the specific predator. 2. The model A simple fox-rabbit Looks-Voltaire model will be used for this experience. For a closed environment with a fixed energy supply, rabbits feed on grass and their population can grow. Without any predators, what will the final population be? If predators are introduced, this predator will reduce the rabbit population, but to what extent? To what extent will this small CEO-system end. Can it be sustained? Will it be chaotic? These questions can be answered by the following analysis.

Scenario 1 (unrestricted growth): In an unrestricted environment, only rabbits are introduced. The birth rate of rabbits is proportional to its population. Hence; positive rate means an increase, otherwise it will means a decrease. Let be the population at time. Then a dynamic model can be built: Let rabbit/day/head, the initial population is 1000. Solve this problem by hand and then via ASSUMING. Plot and record you results. By hand, Absolutely, The block diagram The rabbit and fox populations It is observed that the curve of the above graph is similar to the result of the equation by hand.

Since the rate is positive, the number of rabbit increases continuously. Scenario 2 (Introduction of a predator): Suppose foxes and rabbits are the only animal populations in this environment. Let be the population of foxes at time. Foxes can only hunt rabbits, so foxes will starve if the population of rabbits decline. Let be the mortality rate of foxes, then we have This equation can be solved similar to (1 If the life supporting level is fixed, then when the population of the rabbits grows close to that level, there will be insufficient grass to sustain the population. If the foxes have plentiful supply of abbots, they can breed and grow.

Then we have: However, as the fox population grows, more rabbits will be eaten, and hence there is less food for the foxes. Consequently, the fox population will decline! This inter-relation can be described by the following equation: Solve the behavioral equations (3) and (4). Plot the rabbit and fox populations. Then plot the trajectory of rabbit-fox relationship. By exploring different parameter sets various situations will arise. For instance, either the rabbit or the fox population will grow out of control or diminish; both populations are under a sustained cyclic evolution.

Depict your results on the sample graph and see the overall trajectories. The trajectory of rabbit-fox relationship It is observed that the increment of fox follows by the increment of rabbit. As the rabbits increase, more rabbits will be eaten by fox. Then, the foxes increase and the rabbits decrease. Lastly the foxes also decrease. This phenomenon is repeated periodically. Scenario 3 (Finite resource): A realistic situation is that the environment can only provide a finite amount of resource and therefore an infinite growth situation will never happen.

Now let be the system resource, then the grow rate will be checked by a factor. The dynamic behavior becomes: Solve this equation by ASSUMING with and There are two methods to draw the block diagram After limitation of the growth of rabbit, the growth rate trends to 0. If r(O)=10, the growth factor is positive, so the number of rabbit increases. If the growth factor is negative, so the number of rabbit decreases. Scenario 4 (Both population with finite resource and saturation): A more realistic model can be built by including the system’s resource level.

In addition, as the abbot population grows, there is limit to the amount the foxes can consume and then a saturation in consumption state will be reached. Let be the fox’s eating saturation factor. With all these factors together, we have (b) By selecting various set of parameters, rabbits’ growth rate rabbit consumption by foxes fox’s’ increase rate due to consumption of rabbits. Fox decline rate due to starvation C resource capacity S foxes’ eating saturation factor Using ASSUMING to simulation various scenarios, plot your system behaviors in the form of a time series and state trajectories. Comment on your finding.

As I have mentioned from scenario 2, the population of fox is followed by the population of rabbit. We have also observed the same result from this part. However, the population of rabbit and fox trends certain value in this part. We called this is stable in this system. This result comes from the variable (fox decline rate due to starvation), S (foxes’ eating saturation factor) and C (resource capacity). There is a limitation or saturation of fox eating rabbit and the rabbits have enough resource to maintain their life. Therefore, both the population of rabbits and fox keep constant, rather than trending zero.