Pacific salmon populations have discrete breeding cycles in which they return from the ocean to streams to reproduce and then die. This occurs every one to five years, depending on the species. (a) Suppose that each fish must first survive predation by bears while swimming upstream, and predation occurs with probability d. After swimming upstream, each fish produces b offspring before dying. The stream is then stocked with m additional newly hatched fish before all fish then swim out to sea. What is the discrete-time recursion for the population dynamics, assuming that there is no mortality while at sea? You should count the population immediately before the upstream journey. nt + 1 = (b) Suppose that, instead of preying on fish while they swim upstream, bears do so only while the fish are swimming downstream. What is the discrete-time recursion for the population dynamics? (Again assume there is no mortality while at sea.) nt+1 = (c) Which of the recursions obtained in parts (a) and (b) predicts the largest increase in population size from one year to the next? O recursion formula from part (a) O recursion formula from part (b) Justify your answer both mathematically and in terms of the underlying biology. You can assume that 0 < d < 1 and b > 0.
Pacific salmon populations have discrete breeding cycles in which they return from the ocean to streams to reproduce and then die. This occurs every one to five years, depending on the species. (a) Suppose that each fish must first survive predation by bears while swimming upstream, and predation occurs with probability d. After swimming upstream, each fish produces b offspring before dying. The stream is then stocked with m additional newly hatched fish before all fish then swim out to sea. What is the discrete-time recursion for the population dynamics, assuming that there is no mortality while at sea? You should count the population immediately before the upstream journey. nt + 1 = (b) Suppose that, instead of preying on fish while they swim upstream, bears do so only while the fish are swimming downstream. What is the discrete-time recursion for the population dynamics? (Again assume there is no mortality while at sea.) nt+1 = (c) Which of the recursions obtained in parts (a) and (b) predicts the largest increase in population size from one year to the next? O recursion formula from part (a) O recursion formula from part (b) Justify your answer both mathematically and in terms of the underlying biology. You can assume that 0 < d < 1 and b > 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:Pacific salmon populations have discrete breeding cycles in which they return from the ocean to streams to reproduce and then die. This occurs every one to five years,
depending on the species.
(a) Suppose that each fish must first survive predation by bears while swimming upstream, and predation occurs with probability d. After swimming upstream,
each fish produces b offspring before dying. The stream is then stocked with m additional newly hatched fish before all fish then swim out to sea. What is the
discrete-time recursion for the population dynamics, assuming that there is no mortality while at sea? You should count the population immediately before the
upstream journey.
nt + 1 =
(b) Suppose that, instead of preying on fish while they swim upstream, bears do so only while the fish are swimming downstream. What is the discrete-time
recursion for the population dynamics? (Again assume there is no mortality while at sea.)
nt+1
(c) Which of the recursions obtained in parts (a) and (b) predicts the largest increase in population size from one year to the next?
recursion formula from part (a)
recursion formula from part (b)
Justify your answer both mathematically and in terms of the underlying biology. You can assume that 0 < d < 1 and b > 0.
Expert Solution
Hint:
The difference equation modelling is given by
Nt+1=f(N1) where Nt denotes the population size.
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