Systems and Control
Semester B 03/04
Experiment 1 - System Dynamics and Behavior
Electronic and Communication Engineering
EE3114 Systems and Control
System Dynamics and Behavior
Dynamic systems like dc-servomotors, financial systems, logistic models, internet systems and eco-systems can be described by a set of coupled differential equations. Based on this model, one can study the behavior of such a system under various external factors such as initial conditions, variables’ interrelation changes, stead state responses and stability issues. In this experiment, a simple Loika-Volterra type model was chosen to demonstrate these behaviors. The concept of state trajectory, limit cycles and stability will be introduced.
There is a trophic level controls the dynamics of another trophic level is central to ecology. Energy flows between trophic levels. The input of energy to a higher trophic level population from a lower trophic level population can be controlled by the amount of energy available in the lower level, or by the amount of energy the higher trophic 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 species(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 Loika-Volterra 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 eco-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 SIMULINK. Plot and record you results.
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 rabbits, they can breed and grow. Then we have:
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