Question

# Question 2. (4 points) Intensive harvesting of a population of a fish species can cause population extinction. We consider the following two models for harvesting of a given fish species and analyze how the extinction depends on the nature of the harvesting. The population size P (measured in thousands) is a function of harvesting effort h. (The harvesting effort is a mathematical measure of "fishing effort", which you are not expected to know in details.) \begin{aligned} &\text { Model }\\ &\text { 1: } \quad P(h)=\left\{\begin{array}{ll} 3(1-h) & \text { if } 0 \leq h \leq 1 \\ 0 & \text { if } h>1 \end{array}\right. \end{aligned} \begin{aligned} &\text { Model }\\ &\text { 2: } \quad P(h)=\left\{\begin{array}{ll} 1+\sqrt{4-3 h} & \text { if } 0 \leq h \leq \frac{4}{3} \\ 0 & \text { if } h>\frac{4}{3} \end{array}\right. \end{aligned} (1) (1 point) What is the initial population of this species when no harvesting efforts were applied at all? (2) (1 point) Draw the graph of each model. (3) (2 points) Here you will analyze each model in terms of the Intermediate Value Theorem by answering the following questions. Which model has a situation where a small change in harvesting effort causes a sudden extinction? In Model 2, is there a harvesting effort to obtain the population of 500?  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6