Problem 7 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Show that for $$ p(t)=\frac{M p_{0}}{p_{0}+\left(M-p_{0}\right) e^{-r t}} $$a. \(p(0)=p_{0}\) $$\text { b. } \lim _{t \rightarrow \infty} p(t)=M$$

Step-by-Step Solution

Answer

a. \(p(0)=p_0\); b. \(\lim_{t \to \infty} p(t) = M\).

Step 1: Understand the Function

The function given is \(p(t) = \frac{M p_{0}}{p_{0} + (M-p_{0})e^{-rt}}\). This is a type of logistic function commonly used in modeling population growth, where \(M\) represents the carrying capacity, \(p_0\) is the initial population, and \(r\) is the growth rate.

Step 2: Verify Initial Condition: \(p(0) = p_0\)

To show \(p(0) = p_0\), substitute \(t = 0\) into the function:\[p(0) = \frac{M p_0}{p_0 + (M - p_0)e^{0}}\]Since \(e^{0} = 1\), it simplifies to:\[p(0) = \frac{M p_0}{p_0 + (M - p_0)} = \frac{M p_0}{M} = p_0\]This verifies that when \(t = 0\), \(p(t)\) indeed equals \(p_0\).

Step 3: Find Limit as \(t \rightarrow \infty\)

To determine \(\lim_{t \to \infty} p(t)\), analyze the behavior of \(e^{-rt}\) as \(t\) approaches infinity. Since \(e^{-rt}\) decays to 0, the function simplifies:\[ p(t) = \frac{Mp_0}{p_0 + (M-p_0)\cdot 0} = \frac{Mp_0}{p_0} = M \]This shows that as \(t\) approaches infinity, \(p(t)\) approaches \(M\).

Step 4: Conclusion

The function satisfies both conditions: \(p(0) = p_0\) and \(\lim_{t \to \infty} p(t) = M\). This confirms the behavior of the logistic function as prescribed for population growth.

Key Concepts

Population Growth Model

The logistic function is a popular mathematical tool used to model how populations grow over time. It's like a mathematical blueprint illustrating how a group of living things, like animals or plants, increases in number. The classic equation for this function is:\[p(t) = \frac{M p_0}{p_0 + (M - p_0) e^{-rt}}\]Here, - \( p(t) \) represents the population size at time \( t \), - \( M \) is the maximum number of individuals the environment can support,- \( p_0 \) is the initial population size, and - \( r \) is the growth rate.Unlike a simple exponential model, which assumes unlimited growth, the logistic function shows how the growth rate slows down as the population nears its maximum limit, or carrying capacity. This is because resources like food and space become limited. Notice how the model respects the boundary conditions: when time \( t \) is zero, \( p(t) \) is equal to \( p_0 \), the start size, and as time goes on and approaches infinity, the population size \( p(t) \) levels off, heading towards the carrying capacity \( M \). This attribute makes this function critically important for understanding how real-life populations develop over time.

Exponential Decay

Understanding exponential decay is fundamental to grasping why populations don't grow indefinitely according to the logistic function. The term \( e^{-rt} \) in the function represents the exponential decay factor. When you think of exponential decay, imagine something that is gradually getting smaller, like a melting ice cube or a slowly deflating balloon.In our logistic formula, this component makes sure that as time increases, the influence of the initial conditions, embodied by \( (M-p_0)e^{-rt} \), diminishes. The number \( e^{-rt} \) is simply an exponentiation of \( e \) (approximately 2.718) to the power of a negative number, \(-rt\). As \( t \) grows larger, \(-rt\) becomes more negative, pushing \( e^{-rt} \) closer to zero. This means:

• The effect of the initial population difference from the carrying capacity fades over time, and so does the impact of the growth rate \( r \).
• The decay leads to the stabilization of the population size \( p(t) \).

Eventually, the decaying effect of \( e^{-rt} \) plays a key role in the population not exceeding the carrying capacity \( M \). This is how the growth is naturally curbed, preventing the exponential growth scenario where unchecked populations would continue to expand infinitely.

Carrying Capacity

Carrying capacity is a crucial concept in understanding the limitations of population growth. It symbolizes the maximum population size that an environment or habitat can support indefinitely. When we talk about carrying capacity in the context of the logistic function, we're referring to the parameter \( M \) in the equation.As a population grows, it utilizes resources like food, water, and space. Carrying capacity represents the point at which these resources are completely utilized, and the population cannot continue to grow without exceeding its environment's ability to sustain it. This means:

• Once the carrying capacity is reached, the growth rate slows down and ideally stabilizes, leading to a balanced ecosystem if the conditions remain consistent.
• In the logistic growth model, as time moves forward and exponential decay sets in, the population \( p(t) \) smoothens out approaching \( M \), thus reflecting a state of equilibrium.
• Natural populations often fluctuate around the carrying capacity due to changes in environmental conditions, showing both rises and declines before stabilizing.

This understanding empowers us to predict and manage populations better, whether for wildlife conservation or agricultural planning. It's essential for sustainability as it ensures populations do not grow beyond what their ecosystems can handle.