Which do you think is the best stage-based model to describe the growth of the Goslin population? a) Lefkovitch Matrix model b) Leslie Matrix model c) Exponential model D) logistic model
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Match the following situations with the population growth model that would be most appropriate to use: Growth in a limited environment with continuous A. Exponential reproduction & overlapping generations B. Logistic Growth in a limited environment with discrete reproduction & non-overlapping generations C. Geometric Growth in an unlimited environment with continuous reproduction & overlapping generations_
Niral M.
E4.19 Leslie population model. The Leslie model is used in population ecology to model the changes in a population of organisms over a period of time; see the original reference (Leslie, 1945) and a comprehensive text (Caswell, 2006). In this model, the population is divided into n groups based on age classes; the indices i are ordered increasingly with the age, so that i = 1 is the class of the newborns. The variable x_i(k), i ∈ {1, ..., n}, denotes the number of individuals in the age class i at time k; at every time step k the x_i(k) individuals • produce a number ̑_i x_i(k) of offsprings (i.e., individuals belonging to the first age class), where ̑_i ≥ 0 is a fecundity rate, and • progress to the next age class with a survival rate ̠_i ∈ [0, 1]. If x(k) denotes the vector of individuals at time k, the Leslie population model reads x(k + 1) = Ax(k) = [̑_1 ̑_2 ... ̑_{n-1} ̑_n; ̠_1 0 ... 0 0; 0 ̠_2 ... 0 0; ... ... ... ... ...; 0 0 ... ̠_{n-1} 0] x(k), (E4.4) where A is referred to as the Leslie matrix. Consider the following two independent sets of questions. First, assume ̑_i > 0 for all i ∈ {1, ..., n} and 0 < ̠_i ≤ 1 for all i ∈ {1, ..., n - 1}. (i) Prove that the matrix A is primitive. (ii) Let p_i(k) = x_i(k) / ∑_{i=1}^n x_i(k) denote the percentage of the total population in class i at time k. Call p(k) the population distribution at time k. Compute lim_{k→∑} p(k) as a function of the spectral radius ̡(A) and the parameters (̑_i, ̠_i), i ∈ {1, ..., n}. Hint: Obtain a recursive expression for the components of the right dominant eigenvector of A (iii) Assume ̠_i = ̠ > 0 and ̑_i = ̑/n for i ∈ {1, ..., n}. What percentage of the total population belongs to the eldest class asymptotically, that is, what is lim_{k→∑} p_n(k)? (iv) Find a sufficient condition on the parameters (̑_i, ̠_i), i ∈ {1, ..., n}, so that the population will eventually become extinct. Second, assume ̑_i ≥ 0 for i ∈ {1, ..., n} and 0 ≤ ̠_i ≤ 1 for all i ∈ {1, ..., n - 1}. (v) Find a necessary and sufficient condition on ̑_1, ..., ̑_n, and ̠_1, ..., ̠_{n-1} so that the Leslie matrix A is irreducible. (vi) For an irreducible Leslie matrix (as in the previous point (v)), find a sufficient condition on the parameters (̑_i, ̠_i), i ∈ {1, ..., n}, that ensures that the population will not go extinct.
Supreeta N.
A new weedy plant has invaded the San Joaquin valley from South Africa and is threatening production in our farmland. Because of our mild conditions and heavy irrigation, the weed is able to successfully flower and set seed year round. Which model would be most appropriate for predicting population growth of this weedy species? A. geometric growth model B. exponential growth model C. basic population growth model D. breeder's equation
Adi S.
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