What is the difference between the exponential and logistic models of population growth?

The exponential growth model describes a population that grows at a constant rate proportional to its size. It assumes that there are unlimited resources available for the population to grow. The exponential growth model is represented by the formula: $$ N(t) = N_0 * e^{rt} $$ Where \(N(t)\) is the population size at time \(t\), \(N_0\) is the initial population, \(r\) is the intrinsic growth rate, and \(e\) is the base of the natural logarithm (approximately equal to 2.71828).

The logistic growth model takes into account the limited availability of resources and the fact that population growth slows down as the population reaches its carrying capacity. The logistic growth model is represented by the formula: $$ N(t) = \frac{K * N_0 * e^{rt}}{K + N_0 * (e^{rt} - 1)} $$ Where \(N(t)\) is the population size at time \(t\), \(N_0\) is the initial population, \(r\) is the intrinsic growth rate, \(K\) is the carrying capacity, and \(e\) is the base of the natural logarithm (approximately equal to 2.71828).

The main differences between the exponential and logistic growth models are: 1. Exponential growth assumes unlimited resources, while logistic growth considers limited resources and carrying capacity. 2. Exponential growth leads to a constantly increasing growth rate, whereas logistic growth starts rapidly and then slows down as the population reaches its carrying capacity. 3. Exponential growth results in a J-shaped curve when plotted on a graph, while logistic growth produces an S-shaped curve. This is because exponential growth continuously accelerates, while logistic growth accelerates at first and then decelerates.

Example of exponential growth: A bacteria population in a petri dish, with abundant resources, can experience exponential growth in its early stages. If the initial population is 100 and the intrinsic growth rate is 2 per hour, after 2 hours the population would be: $$ N(2) = 100 * e^{2 * 2} = 100 * e^4 \approx 2{,}684 $$ Example of logistic growth: A population of rabbits on an island with limited resources, such as food and shelter, will follow the logistic growth model. If the initial population is 100 rabbits, the intrinsic growth rate is 0.2 per year, the carrying capacity is 1{,}000 rabbits, after 5 years the population would be: $$ N(5) = \frac{1000 * 100 * e^{0.2 * 5}}{1000 + 100 * (e^{0.2 * 5} - 1)} \approx 388 $$