4. (**) The populations of two types of fish obey the discrete-time dynamical system $\frac{4}{5}a_n - \frac{1}{5}b_n$, $a_{n+1} = $ $b_{n+1} = -\frac{2}{5}a_n + \frac{3}{5}b_n$ (a) (2 marks) Write the dynamical system in the form $v_{n+1} = Av_n$ where $v_n = (a_n, b_n)^T$, and A is a matrix which you should specify. (b) (8 marks) Determine matrices P and D such that D is diagonal, and $A = PDP^{-1}$. (c) (7 marks) Find explicit expressions for $a_n$ and $b_n$ subject to the initial conditions $a_0 = 500$ and $b_0 = 300$. (d) (3 marks) Find the long-time behaviour of the populations in the limit $n \to \infty$.
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Let's define the vector u = [an, bn] and the matrix A as: A = [[4, 2], [3, 10]] Now we can rewrite the dynamical system as: u(n+1) = Au(n) (b) To determine matrices P and D such that D is diagonal and APDPT, we need to find the eigenvalues and Show more…
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Fish Population Growth The general discrete dynamical system model for a fish population assumes a per capita production rate given by p(N) := r / (1 + 0.1N), where N is the number of fish in tens of thousands (i.e., N = 5 means 50,000 fish). The value of r, the maximal possible reproduction rate, depends on the particular species of fish and environmental factors. The discrete-time dynamical system that describes the development of a fish population Nt is given by: Nt+1 = (r / (1 + 0.1Nt)) * Nt Assume that the unit for t is years. That is, t = 1 means one year. 1. Suppose that for Atlantic cod, r = 2.5. What are the equilibrium points in this case? (Solve for your answer algebraically.) 2. Determine the stability of each equilibrium point found in the previous part, using the derivative test.
Adi S.
The size of an undisturbed fish population has been modeled by the formula $$ p_{n+1}=\frac{b p_{n}}{a+p_{n}} $$ where $p_{n}$ is the fish population after $n$ years and $a$ and $b$ are positive constants that depend on the species and its environment. Suppose that the population in year 0 is $p_{0} > 0$ . (a) Show that if $\left\{p_{n}\right\}$ is convergent, then the only possible values for its limit are 0 and $b-a$ . (b) Show that $p_{n+1}<(b / a) p_{n}$ (c) Use part (b) to show that if $a > b,$ then $\lim _{n \rightarrow \infty} p_{n}=0$ in other words, the population dies out. (d) Now assume that $a < b .$ Show that if $p_{0} < b-a$ , then $\left\{p_{n}\right\}$ is increasing and $0 < p_{n}< b-a$ . Show also that if $p_{0} >b-a$ , then $\left\{p_{n}\right\}$ is decreasing and $p_{n} > b-a$ Deduce that if $a < b$ , then $\lim _{n \rightarrow \infty} p_{n}=b-a$
Infinite Sequences and Series
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The size of an undisturbed fish population has been modeled by the formula $ p_{n + 1} = \frac {bp_n}{a + p_n} $ where $ p_n $ is the fish population after $ n $ years and $ a $ and $ b $ are positive constants that depend on the species and its environment. Suppose that the population in year 0 is $ p_0 > 0. $ (a) Show that if $ \{ p_n \} $ is convergent, then the only possible values for its limit are 0 and $ b - a $. (b) Show that $ p_{n + 1} < (b/a)p_n $. (c) Use part (b) to show that if $ a > b, $ then $ \lim_{n \to \infty} p_n = 0 $; in other words, the population dies out. (d) Now assume that $ a < b $. Show that if $ p_0 < b - a $, then $ \{ p_n \} $ is increasing and $ 0 < p_n < b - a $. Show also that if $ p_0 > b - a $, then $ \{ p_n \} $ is decreasing and $ p_n > b - a $. Deduce that if $ a < b $, then $ \lim_{n \to \infty} p_n = b - a $.
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