Final answer:
The Gompertz growth model with harvesting shows that a threshold population size M exists where growth transitions from positive to negative due to the balance between the natural logarithm term and the harvesting term in the model's differential equation.
Step-by-step explanation:
The Gompertz growth model is a mathematical model for a time series, where growth is slowest at the start and end of a time period. In the modified version of the Gompertz model with harvesting, the rate of change of the population P(t) is given by the differential equation \(\frac{{dP}}{{dt}} = P \ln(L/P) - qEP\), where qE is the harvesting term, L is a constant representing the upper limit of the population size, and P(t) is the population at time t. To prove that there is a number M for which P(t) is increasing when 0 < P(t) < M and decreasing when P(t) > M, we need to analyze the right-hand side of the equation for sign changes.
For a population size P(t) below the carrying capacity L, the term \(\ln(L/P)\) is positive, and if we assume that P(t) is sufficiently small so that \(P \ln(L/P) > qEP\), then the change in population \(\frac{{dP}}{{dt}}\) is positive, leading to growth. However, as P(t) increases and approaches M, the term \(P \ln(L/P)\) diminishes, eventually becoming equal to the harvesting term \(qEP\), making \(\frac{{dP}}{{dt}}\) equal to zero. This is the threshold M, where the population stops growing. Beyond this point, for P(t) > M, \(P \ln(L/P)\) becomes smaller than \(qEP\), and \(\frac{{dP}}{{dt}}\) becomes negative, leading to a decline in population.