Basic population model: a population of rabbits can be divided juveniles and adu

Question

Basic population model: a population of rabbits can be divided juveniles and adults. It takes rabbits about 1 year to mature to adulthood. On average, each breeding pair has a litter of 4 babies (2 babies per adult) with about 60% of babies surviving to adulthood. The death rate of adults is approximately 15% of the population each year. Let the state of this system be the populations of juveniles and adults in year t:.........

Discrete Time Systems Suppose there is some system (physical, economic, chemical, biological, etc.), and in this system there are measurable quantities 31x2,..., ^n Suppose these quantities are measured at time t. We can put these quantities into a vector and call the vector the state of the system at time t: x1 Suppose we want to determine the state of the system one time-unit in the future, given we know the current state. For example, to determine the state one year from now given we know the current state. In vector notation, we want to get X1+1 from X.. Often the transition from X, to Xi+1 takes the form of matrix multiplication,

Explanation / Answer

Code in MATLAB

value = 0;

A = [0 2;0.6 0.85];

x0 =[0;100];

x1 = round(A*x0);

x2 = round(A*x1);

x3 = round(A*x2);

x4 = round(A*x3);

r = x0(2)/x0(1);

adult = [0,0,0,0,0,0,0];

i = 1;

while abs(r - value)>0.001

  

adult(i) = x0(2);

r = x0(2)/x0(1);

y = round(A*x0);

x0 = y;

disp(x0(2));

value = x0(2)/x0(1);

i = i+1;

end

y0 =[0;0];

y1 = round(A*y0);

y2 = round(A*y1);

y3 = round(A*y2);

y4 = round(A*y3);

(a)

>> x1

x1 =

200
85

>> x2

x2 =

170
192

>> x3

x3 =

384
265

>> x4

x4 =

530
456

(b) Number of Adults array =

100 85 192 265 456 706 1147 1822 2925 4673 7482

After 11 years number of adults crosses 5000.

(c) yes it coverges to 0.8 (found using code in MATLAB)

(d) if X0 = 0 , X1 = 0 and X2 = 0 and so on every Xn = 0 i.e. if there are no adults or juveniles in the start , no adult or juveniles will be reproduced in the future

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