[Solved]Please Solve Python Exercise82 Thelotka Volterra Equations Lotka Volterra Equations Mathem Q37088202
*Please solve in Python* Exercise8.2: TheLotka–Volterraequations
The Lotka–Volterra equations are a mathematical model ofpredator–prey interactions between biological species. Let twovariables x and y be proportional to the size of the populations oftwo species, traditionally called “rabbits” (the prey) and “foxes”(the predators). You could think of x and y as being the populationin thousands, say, so that x= 2 means there are 2000 rabbits.Strictly the only allowed values of x and y would then be multiplesof 0.001, since you can only have whole numbers of rabbits orfoxes. But 0.001 is a pretty close spacing of values, so it’s adecent approximation to treat x and y as continuous real numbers solong as neither gets very close to zero.
In the Lotka–Volterra model the rabbits reproduce at a rateproportional to their population but are eaten by the foxes at arate proportional to both their own population and the populationof foxes: dx/dt =αx−βxy, where α and β are constants. At the sametime, the foxes reproduce at a rate proportional the rate at whichthey eat rabbits—because they need food to grow and reproduce—butalso die of old age at a rate proportional to their own population:dy/dt =γxy−δy,where γ and δ are also constants.
a) Write a program to solve these equations using thefourth-order Runge–Kutta method for the case α = 1, β =γ = 0.5, andδ = 2, starting from the initial condition x = y = 2. Have theprogram make a graph showing both x and y as a function of time onthe same axes fromt= 0 tot= 30. (Hint: Notice that the differentialequations, in this case, do not depend explicitly on timet—invector notation, the right-hand side of each equation is a functionf(r) with no t dependence. You may nonetheless find it convenient todefine a Python function f(r,t) including the time variable. Youdon’t have to do it that way, but it can avoid some confusion.
We were unable to transcribe this imageExercise 8.2: The Lotka-Volterra equations The Lotka-Volterra equations are a mathematical model of predator-prey interactions between biological species. Let two variables x and y be proportional to the size of the populations of two species, traditionally called “rabbits” (the prey) and “foxes” (the predators). You could think of x and y as being the population in thousands, say, so that x 2 means there are 2000 rabbits. Strictly the only allowed values of x and y would then be multiples of 0.001, since you can only have whole numbers of rabbits or foxes. But 0.001 is a pretty close spacing of values, so it’s a decent approximation to treat x and y as continuous real numbers so long as neither gets very close to zero In the Lotka-Volterra model the rabbits reproduce at a rate proportional to their popula tion, but are eaten by the foxes at a rate proportional to both their own population and the population of foxes where α and β are constants. At the same time the foxes reproduce at a rate proportional the rate at which they eat rabbits-because they need food to grow and reproduce-but also die of old age at a rate proportional to their own population: di dt where γ and δ are also constants. a) Write a program to solve these equations using the fourth-order Runge-Kutta method for the case a 1, β γ-0.5, and δ-2, starting from the initial condition x-y-2. Have the program make a graph showing both x and y as a function of time on the same axes from t 0to 30. (Hint: Notice that the differential equations in this case do not depend explicitly on time t-in vector notation, the right-hand side of each equation is a function f(r) with no t dependence. You may nonetheless find it convenient to define a Python function f(r,t) including the time variable, so that your program takes the same form as programs given earlier in this chapter. You don’t have to do it that way, but it can Show transcribed image text
Exercise 8.2: The Lotka-Volterra equations The Lotka-Volterra equations are a mathematical model of predator-prey interactions between biological species. Let two variables x and y be proportional to the size of the populations of two species, traditionally called “rabbits” (the prey) and “foxes” (the predators). You could think of x and y as being the population in thousands, say, so that x 2 means there are 2000 rabbits. Strictly the only allowed values of x and y would then be multiples of 0.001, since you can only have whole numbers of rabbits or foxes. But 0.001 is a pretty close spacing of values, so it’s a decent approximation to treat x and y as continuous real numbers so long as neither gets very close to zero In the Lotka-Volterra model the rabbits reproduce at a rate proportional to their popula tion, but are eaten by the foxes at a rate proportional to both their own population and the population of foxes where α and β are constants. At the same time the foxes reproduce at a rate proportional the rate at which they eat rabbits-because they need food to grow and reproduce-but also die of old age at a rate proportional to their own population: di dt where γ and δ are also constants. a) Write a program to solve these equations using the fourth-order Runge-Kutta method for the case a 1, β γ-0.5, and δ-2, starting from the initial condition x-y-2. Have the program make a graph showing both x and y as a function of time on the same axes from t 0to 30. (Hint: Notice that the differential equations in this case do not depend explicitly on time t-in vector notation, the right-hand side of each equation is a function f(r) with no t dependence. You may nonetheless find it convenient to define a Python function f(r,t) including the time variable, so that your program takes the same form as programs given earlier in this chapter. You don’t have to do it that way, but it can
Expert Answer
Answer to *Please solve in Python* Exercise8.2: TheLotka–Volterra equations The Lotka–Volterra equations are a mathematical mo… . . .
OR
