In the following equations A) dN/dt = rN B) Nt = Noert C) dN/dt = rN(K-N÷k) dN/dt = rN × N/K Exponential population growth is described by 1) a and b 2) a only 3) c only 4) b and d
The Correct Answer and Explanation is:
The correct answer is 1) a and b.
Explanation:
The given equations represent different models of population growth, specifically looking at exponential and logistic growth.
- Equation A: dN/dt = rN This equation is a basic form of the exponential growth model. Here, dN/dt represents the rate of change of the population over time, r is the growth rate, and N is the population size. This equation shows that the population grows at a constant rate r and the rate of change is directly proportional to the current population. As long as the population grows exponentially without constraints, this equation holds. Hence, equation A represents exponential growth.
- Equation B: Nt = Noe^rt This equation is another way of writing the exponential growth model. Nt is the population at time t, No is the initial population size, e is the base of the natural logarithm, and r is the growth rate. This is the solution to the differential equation in equation A, describing population growth over time. It clearly shows exponential growth, where the population increases without any environmental limits or resource constraints. Therefore, equation B also represents exponential growth.
- Equation C: dN/dt = rN(K – N) / K This is the logistic growth model, which takes into account carrying capacity (K) of the environment. Here, the growth rate slows as the population approaches the carrying capacity. Initially, when the population is small, it grows exponentially (similar to equation A), but as the population increases, the growth rate slows due to resource limitations. This is not exponential growth, so equation C does not represent exponential growth.
- Equation D: dN/dt = rN × N / K This equation is similar to equation C but slightly different in form. It represents a type of logistic growth with a focus on limiting factors. While it may look similar to exponential growth initially, it incorporates a population density-dependent limiting factor. Therefore, it is not purely exponential growth.
Thus, equations A and B represent exponential growth models, making option 1) the correct answer.
