CoCalc Public Filesprojects_F20 / dds_Proj11 / Population Model.ipynbOpen with one click!
Author: Julia Burnside
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Compute Environment: Ubuntu 20.04 (Default)

A Population Model

The average rat can live up to 4 years. Below is a table of birth and survival rates for rats observed at the local park.

Age (Years) Birth Rate Survival Rate Population in 2019
0-1 0.000 0.845 61
1-2 0.450 0.975 75
2-3 0.391 0.965 64
3-4 0.264 0.012 43
Construct a discrete population model for this population of rats.

First we need to separate the population into 4 age groups;

An=(01) age population,A_n=(0-1)\ age\ population,

Bn=(12) age population,B_n=(1-2)\ age\ population,

Cn=(23) age population,C_n=(2-3)\ age\ population,

Dn=(34) age population.D_n=(3-4)\ age\ population.

Our initial population values as of 2019 are;

A0=61,A_0=61,

B0=75,B_0=75,

C0=64,C_0=64,

D0=43.D_0=43.

The birth rate each year for each group must be added to A;

An+1=0.450Bn+0.391Cn+0.264DnA_{n+1}=0.450 \cdot B_n + 0.391 \cdot C_n + 0.264 \cdot D_n

Each year, a percentage of the population of each age group graduates to the next group, so we must transfer the survival rate of the previous age group to the subsequent population group;

Bn+1=0.845An,B_{n+1}=0.845 \cdot A_n,

Cn+1=0.975Bn,C_{n+1}=0.975 \cdot B_n,

Dn+1=0.965Cn.D_{n+1}=0.965 \cdot C_n.

In [39]:
import numpy as np import matplotlib.pyplot as plt def rat(n): A=[61] B=[75] C=[64] D=[43] for i in range(n): A_old=A[-1] B_old=B[-1] C_old=C[-1] D_old=D[-1] A_new=0.45*B_old+0.391*C_old+0.264*D_old B_new=0.845*A_old C_new=0.975*B_old D_new=0.965*C_old A.append(A_new) B.append(B_new) C.append(C_new) D.append(D_new) return A, B, C, D Y1,Y2,Y3,Y4=rat(6) plt.plot(Y1, label="Age 0-1") plt.plot(Y2, label="Age 1-2") plt.plot(Y3, label="Age 2-3") plt.plot(Y4, label="Age 3-4") plt.xlabel("2019-2025 (Six Year Period)") plt.ylabel("Rat Population") plt.legend() plt.show()
In [42]:
rat(6) rat(20) rat(200)
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Estimate the population of rats at the local part in 2025.

Assuming the population model is correct, the rat population for all age groups will decrease between 2019-2025. If we continue to monitor this trend over a long enough period of time, the population will completely bottom out, suggesting the birth rate vs the survival rate is not sustainable.

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