un ejercicio
Publicado por german (1 intervención) el 05/11/2008 23:00:38
tengo que hacer un ejercicio para una materia de la facultad pero el profesor es pesimo y no entiendo nada sobre programar en matlab, si alguien me lo puede resolver le agradesco mucho.
23. In the 1920’s, the Italian mathematician Umberto Volterra proposed the following mathematical model of a predator-prey situation to explain why, during the firstWorldWar, a larger percentage of the catch of Italian fishermen consisted of sharks and other fish eating fish than was true both before and after the war. Let x(t) denote the population of the prey, and let y(t) denote the population of the predators.
In the absence of the predators, the prey population would have a birth rate greater than its death rate, and consequently would grow according to the exponential model of population growth, i.e. the growth rate of the population would be proportional to the population itself. The presence of the predator population has the effect of reducing the growth rate, and this reduction depends on the number of encounters between individuals of the two species. Since it is reasonable to assume that the number of such encounters is proportional to the number of individuals of each population, the reduction in the growth rate is also proportional to the product of the two populations, i.e., there are constants a and b such that
x = a x − b x y. (8.24)
Since the predator population depends on the prey population for its food supply it is natural to assume that in the absence of the prey population, the predator population would actually decrease, i.e. the growth rate would be negative. Furthermore the (negative) growth rate is proportional to the population. The presence of the prey population would provide a source of food, so it would increase the growth rate of the predator species. By the same reasoning used for the prey species, this increase would be proportional to the product of the two populations. Thus, there are constants c and d such that
y = −c y + d x y. (8.25)
a) A typical example would be with the constants given by a = 0.4, b = 0.01, c = 0.3, and d = 0.005.
Start with initial conditions x1(0) = 50 and x2(0) = 30, and compute the solution to (8.24) and (8.25) over the interval [0, 100]. Prepare both a time plot and a phase plane plot.
AfterVolterra had obtained his model of the predator-prey populations, he improved it to include the effect of “fishing,” or more generally of a removal of individuals of the two populations which does not discriminate between the two species. The effect would be a reduction in the growth rate for each of the populations by an amount which is proportional to the individual populations. Furthermore, if the removal is truly indiscriminate, the proportionality constant will be the same in each case. Thus, the model in equations (8.24) and (8.25) must be changed to
x =a x − b x y − e x,
y =−c y + d x y − e y,
(8.26)
where e is another constant.
b) To see the effect of indiscriminate reduction, compute the solutions to the system in (8.24) when e = 0, 0.01, 0.02, 0.03, and 0.04, and the other constants are the same as they were in part a). Plot the five solutions on the same phase plane, and label them properly.
c) Can you use the plot you constructed in part b) to explain why the fishermen caught more sharks duringWorldWar I?You can assume that because of the war they did less fishing.
23. In the 1920’s, the Italian mathematician Umberto Volterra proposed the following mathematical model of a predator-prey situation to explain why, during the firstWorldWar, a larger percentage of the catch of Italian fishermen consisted of sharks and other fish eating fish than was true both before and after the war. Let x(t) denote the population of the prey, and let y(t) denote the population of the predators.
In the absence of the predators, the prey population would have a birth rate greater than its death rate, and consequently would grow according to the exponential model of population growth, i.e. the growth rate of the population would be proportional to the population itself. The presence of the predator population has the effect of reducing the growth rate, and this reduction depends on the number of encounters between individuals of the two species. Since it is reasonable to assume that the number of such encounters is proportional to the number of individuals of each population, the reduction in the growth rate is also proportional to the product of the two populations, i.e., there are constants a and b such that
x = a x − b x y. (8.24)
Since the predator population depends on the prey population for its food supply it is natural to assume that in the absence of the prey population, the predator population would actually decrease, i.e. the growth rate would be negative. Furthermore the (negative) growth rate is proportional to the population. The presence of the prey population would provide a source of food, so it would increase the growth rate of the predator species. By the same reasoning used for the prey species, this increase would be proportional to the product of the two populations. Thus, there are constants c and d such that
y = −c y + d x y. (8.25)
a) A typical example would be with the constants given by a = 0.4, b = 0.01, c = 0.3, and d = 0.005.
Start with initial conditions x1(0) = 50 and x2(0) = 30, and compute the solution to (8.24) and (8.25) over the interval [0, 100]. Prepare both a time plot and a phase plane plot.
AfterVolterra had obtained his model of the predator-prey populations, he improved it to include the effect of “fishing,” or more generally of a removal of individuals of the two populations which does not discriminate between the two species. The effect would be a reduction in the growth rate for each of the populations by an amount which is proportional to the individual populations. Furthermore, if the removal is truly indiscriminate, the proportionality constant will be the same in each case. Thus, the model in equations (8.24) and (8.25) must be changed to
x =a x − b x y − e x,
y =−c y + d x y − e y,
(8.26)
where e is another constant.
b) To see the effect of indiscriminate reduction, compute the solutions to the system in (8.24) when e = 0, 0.01, 0.02, 0.03, and 0.04, and the other constants are the same as they were in part a). Plot the five solutions on the same phase plane, and label them properly.
c) Can you use the plot you constructed in part b) to explain why the fishermen caught more sharks duringWorldWar I?You can assume that because of the war they did less fishing.
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