To solve this, we will model the data using the logistic growth equation:

$N(t) = \frac{K}{1 + Ae^{-Bt}}$

where:

• $K$ is the carrying capacity,
• $A$ and $B$ are parameters determined from the data,
• $t$ is time, and $N(t)$ is the population size at time $t$.

Let me compute the solution step by step.import numpy as np from scipy.optimize import curve_fit import pandas as pd

Define logistic function

def logistic(t, K, A, B): return K / (1 + A * np.exp(-B * t))

Given data

t = np.array([5, 10, 15, 20, 25, 30]) N = np.array([302, 712, 1262, 1832, 2222, 2263])

Fit the logistic model

params, _ = curve_fit(logistic, t, N, maxfev=10000)

Extract parameters

K, A, B = params

Calculate time for N = 1625

def time_for_population(N_target, K, A, B): return np.log((K / N_target - 1) / A) / -B

t_1625 = time_for_population(1625, K, A, B)

K, t_1625/home/sandbox/.local/lib/python3.11/site-packages/scipy/optimize/_minpack_py.py:881: OptimizeWarning: Covariance of the parameters could not be estimated warnings.warn('Covariance of the parameters could not be estimated', (1432.166666588998, 0.023534837441241237)(1432.166666588998, 0.023534837441241237)successHere are the results based on the logistic regression model:

(a) The carrying capacity ($K$) for this population is approximately 1432.17.

(b) The time ($t$) when the population reaches 1625 is approximately 0.02.

Would you like further details or steps explained?

Here are 5 questions to explore further:

1. Would you like to understand how the logistic model parameters were derived?
2. Do you need a graph of the fitted logistic curve and the data points?
3. Should we explore how the carrying capacity affects the model?
4. Would you like an explanation of why logistic regression is suitable for this data?
5. Would you like to verify other population thresholds using the fitted model?

Tip: Logistic regression assumes a sigmoidal growth trend, making it suitable for modeling populations approaching a carrying capacity.