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Problem 3. The third type of functional response is a sigmoidal curve that has predators foraging inefficiently at low prey densities, or that the rate of search for pray increase with pray densities. The equation for this model is V ( N ) = - CN2 a 2 + N2 a) Find the equation that describes the instantaneous rate of predation for this model. Usec = 50 and a = 40. b) Does c mean the same for type II and type III functional responses? Problem 4. The following graph shows the three different type of responses. The first row shows the responses, the second row shows the first derivative of the responses. Identify which curves correspond with which type of response for the graph by labeling curves, A, B, and C with &quot;Response 1&quot;, &quot;Response 2&quot; or &quot;Response 3&quot;. Then circle the maximum rate of predation for each response in each graph. 200 400 200 400 200 400 200 400 200 400 200 400 Bonus : The relationship between the number of species in a genus, x and the number of genera y comprising x species is given by xyl = k where a and k are constants. Find dy

Predation is one of the prime movers of energy through food chains. It is an important factor in the ecology of populations, and is also an important evolutionary force: natural selection favors more effective predators and more evasive prey. Holling (1959) described three different functional responses curves that describes the relationship between the number of prey eaten by a predator, and the prey density, which are used when modeling predator-prey interactions. Problem 1. The first type of functional response is a simple linear relationship between the num- ber of prey eaten per predator, and the prey density. The equation that represents this type of response is V ( N ) = CN + a a) What would be the rate of predation for this type of functional response? b) What is this model assuming? Give an example in where this response could be apply. Problem 2. The second type of functional response is an hyperbolic function that saturates, because it take in account the time it takes to handle the prey (catch, subdue, and consume an individual prey). The equation that represents this type of response is: V ( N ) = - CN a +N where a represents the &quot;handling time&quot;. (a) What does c represent in this equation? (analyze the limit of this function when N -+ 0o). (b) Write the equation for the instantaneous rate of predation for this model when c = 50 and a = 40. (c) Give an example where this type of response could be apply.