Introduction

The dynamics of cell populations are fundamental to understanding various biological processes, from the spread of bacterial infections to the growth of cancerous tumors. At the intersection of biophysics and mathematical biology, studying these dynamics provides insights into how populations of cells grow, divide, and die under different environmental conditions. This knowledge has profound implications for medicine, biotechnology, and ecology.

In biological systems, cells constantly interact with their environment, which influences their division and death rates. For example, bacteria exposed to antibiotics experience changes in their proliferation and survival rates, and cancer cells undergoing treatment are similarly affected by the drugs they encounter. Both scenarios exemplify how external factors, whether steady or fluctuating, can shape the fate of a population. To effectively intervene—whether to reduce harmful cell populations or enhance beneficial ones—we must understand how these environmental factors influence growth and extinction.

Why Study Population Dynamics?

The biophysics of population dynamics involves understanding how individual cell behaviors contribute to overall population trends. Two key processes—cell division and cell death—determine whether a population grows, stabilizes, or declines. In real-world scenarios, these processes are influenced by the environment in time-dependent and time-independent ways. Examples include:

1. Antibiotics and Bacterial Growth: Antibiotic levels in the body vary over time, creating a fluctuating environment that affects bacterial division and death rates. Understanding these effects can help optimize dosage and timing to suppress bacterial populations effectively.

2. Cancer Treatments: Cancer cells often exhibit varying sensitivity to treatment. Drugs may temporarily reduce division rates or increase death rates, but their effectiveness can diminish over time. Studying how these fluctuations affect population dynamics can inform strategies to maximize therapeutic outcomes.

3. Biotechnological Applications: In synthetic biology and tissue engineering, controlling cell population growth is critical. Whether growing cell cultures or managing engineered organisms, it is essential to predict and guide population behaviors under varying conditions.

Objectives of This Exemplar

This exemplar focuses on simulating and analyzing cell population dynamics under two key scenarios:

1. Time-Independent Environment: In this scenario, the division rate $\gamma(\tau)$ and death rate $\delta(\tau)$ depend only on the age of cells but remain constant over time. We study the population's growth rate and the probability of extinction using mathematical techniques like separation of variables and numerical simulations.

2. Time-Dependent Environment: Here, division and death rates fluctuate over time, modeling realistic environments where external factors like drug concentration or nutrient availability change periodically. We examine the relationship between growth rate and the period of these fluctuations, comparing the results to time-independent cases.

Through these simulations, we aim to address critical questions in cell population biophysics: - How do time-dependent environments influence the growth or extinction of a population compared to time-independent ones? - What are the long-term growth rates under different environmental conditions? - Which treatment strategies are most effective for controlling harmful cell populations, such as bacteria or cancer cells?