Problem 2 - Advanced Engineering Mathematics

Question

The population model given in (1) fails to take death into consideration; the growth rate equals the birth rate. In another model of a changing population of a community, it is assumed that the rate at which the population changes is a net rate - that is, the difference between the rate of births and the rate of deaths in the community. Determine a model for the population \(P(t)\) if both the birth rate and the death rate are proportional to the population present at time \(t\).

Step-by-Step Solution

Answer

The population model is \( P(t) = P_0 e^{(b-d)t} \).

Step 1: Define Population Change Model

Let's denote the birth rate constant as \( b \) and the death rate constant as \( d \). Since both rates are proportional to the population at time \( t \), we know the birth rate is \( bP(t) \) and the death rate is \( dP(t) \). Thus, the net rate of change in the population is the difference between these two rates: \( bP(t) - dP(t) \).

Step 2: Formulate the Differential Equation

The net rate of change in population results in the differential equation for the population model, which can be expressed as \( \frac{dP}{dt} = (b - d)P(t) \). This equation captures how the population changes based on the net growth rate \( (b-d) \).

Step 3: Solve the Differential Equation

To solve this differential equation \( \frac{dP}{dt} = (b - d)P(t) \), we separate variables: \( \frac{1}{P} dP = (b-d) dt \). Integrating both sides results in \( \ln |P| = (b-d)t + C \), where \( C \) is the integration constant.

Step 4: Exponentiate to Solve for P(t)

Solve for \( P(t) \) by exponentiating both sides: \( P(t) = e^{(b-d)t + C} = e^C e^{(b-d)t} \). Let \( P_0 = e^C \), representing the initial population. Then, \( P(t) = P_0 e^{(b-d)t} \).

Step 5: Interpret the Model

The model \( P(t) = P_0 e^{(b-d)t} \) describes an exponential growth or decay depending on whether the net rate \( (b-d) \) is positive or negative. If \( b > d \), the population grows; if \( b < d \), the population decreases.

Key Concepts

Population Dynamics

Understanding population dynamics is crucial when studying how a population changes over time. In mathematics, we often use models to represent these changes, especially for populations in biological communities or ecosystems.
A model helps us simulate real-world scenarios with mathematical equations. This is done by incorporating factors like birth and death rates. Population dynamics give a detailed insight into the factors that drive the increase or decrease in population numbers.
We can include factors that influence growth, such as food availability, predation, and diseases. In the model from our exercise, the key idea is to focus on the net change in population, determined by comparing birth rates to death rates. Such models help us visualize and predict future changes, which are vital for planning resources and understanding ecological impacts.

Exponential Growth

Exponential growth occurs when the growth rate of a population is proportional to its current size. In simpler terms, the larger the population, the faster it grows.
This happens because each individual in the population can reproduce, leading to a doubling effect over time. In the differential equation \[\frac{dP}{dt} = (b - d)P(t) \],exponential growth is observed when \( b > d \). The exponential factor arises from \( e^{(b-d)t} \), indicating that population size changes rapidly as time progresses.

• If the growth rate is positive (more births than deaths), we see exponential growth.
• A negative growth rate (more deaths than births) leads to exponential decay.

Such growth has real-world implications for resources and environments as it can lead to rapid overpopulation if unchecked.

Birth and Death Rates

Birth and death rates are central to the study of any population. They determine the relative increase or decrease in population size.
In the model given in the exercise, both rates are assumed to be proportional to the population size at any given time, represented as \( bP(t) \) and \( dP(t) \).Here are a few key points:

• The birth rate \( bP(t) \) signifies the rate at which new individuals are added to the population over time.
• The death rate \( dP(t) \) indicates the rate at which individuals are removed from the population.

By focusing on differences between these rates, we can develop a net rate, \( (b-d)P(t) \).This net rate shows us how fast a population is expected to grow or decline.
By understanding these rates, ecologists and researchers can make informed decisions, helping to manage and conserve species in various ecosystems.