I have a question using Maple program to solve a modeling with differential equa

Question

I have a question using Maple program to solve a modeling with differential equations.

The question has multiple parts:

Certain bacteria population has as model the initial-value problem

a) Doubling time is the time needed for a quantity to double in size. For example, if the doubling time is 3 years, the quantity will double in size every 3 years. Does this model have a doubling time? Explain your answer.

b) What is the long-term behavior of the bacteria population?

c) Plot the solution of the initial value problem over an appropriate interval of time. Make sure your plot includes the horizontal asymptote obtained given by the long term behavior of the solution.

I need to use maple commands to solve this problem. I appreciate your time for help. Thank you!

Explanation / Answer

with(plots): with(DEtools): with(plottools):with(LinearAlgebra): with(Statistics):

v1:=1: f:=-4: v2:=2.515: omega:=1: epsilon:=0.001: k:=0:
sys:=diff(u1(t),t)=v1*u1(t)-(omega+k*u2(t)^2)*u2(t)-(u1(t)^2+u2(t)^2+3*z(t)^2)*u1(t),
diff(u2(t),t)=(omega+k*u1(t)^2)*u1(t)+v1*u2(t)-(u1(t)^2+u2(t)^2+3*z(t)^2)*u2(t),
diff(z(t),t)=z(t)*(-v1+3*u1(t)^2+3*u2(t)^2+z(t)^2)+epsilon*z(t)*(v2+f*z(t)^4):

t_start:=50: t_end:=300: dt:=0.05: fs:=1/dt:

solA:=dsolve(,
,
type=numeric, method=rkf45, maxfun=0,
output=Array([seq(i,i=t_start..t_end, dt)])):

u1:=solA[2,1][..,2]:
u2:=solA[2,1][..,3]:
z:=solA[2,1][..,4]:

u0:=sqrt~(u1^~2+u2^~2+z^~2):

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