Population Model.ipynb

Kernel: Python 3 (system-wide)

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;

$A_n=(0-1)\ age\ population,$

$B_n=(1-2)\ age\ population,$

$C_n=(2-3)\ age\ population,$

$D_n=(3-4)\ age\ population.$

Our initial population values as of 2019 are;

$A_0=61,$

$B_0=75,$

$C_0=64,$

$D_0=43.$

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

$A_{n+1}=0.450 \cdot B_n + 0.391 \cdot C_n + 0.264 \cdot D_n$

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;

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

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

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

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()

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.