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...State Infant mort 99 CDR HS drop low BW Teen Births Teen Deaths Alabama 9.8 49 10 9.3 38 84 Alaska 5.7 21 8 5.8 25 69 Arizona 6.8 21 17 6.9 42 61 Arkansas 8 46 12 8.6 38 73 California 5.4 7 9 6.1 31 41 Colorado 6.7 18 14 8.3 29 58 Connecticut 6.1 3 8...
...Year Rain (in) 1980 8.96 1981 10.71 1982 31.28 1983 10.43 1984 12.82 1985 17.86 1986 7.66 1987 12.48 1988 8.08 1989 7.35 1990 11.99 1991 21 1992 27.36 1993 8.14 1994 24.35 1995 12.46 1996 12.4 1997 31.01 1998 9.09 1999 11.57 2000 17.94 2001 4.42
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...mortality calcium 1702 44 1309 59 1259 133 1427 27 1724 6 1175 107 1486 5 1456 90 1696 6 1236 101 1711 13 1444 14 1591 49 1987 8 1495 14 1369 68 1257 50 1587 75 1713 71 1557 13 1640 57 1709 71 1625 20 1527 60 1627 53 1486 122 1485 81 1378 71 1519 21 ...
...US Models Others 16.9 16.2 15.5 20.3 19.2 31.5 18.5 30.5 30 21.5 30.9 31.9 20.6 37.3 20.8 27.5 18.6 27.2 18.1 34.1 17 35.1 17.6 29.5 16.5 31.8 18.2 22 26.5 17 21.9 21.6 27.4 28.4 28.8 26.8 33.5 34.2
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Stochastic 5.7 models Our treatment of age-structured models thus far has been deterministic. An alternative approach to density dependence is to introduce stochastic effects to regulate population size. [It is important to note that although the px in a Leslie matrix model are survival probabilities, fixed values for these are used rather than any distribution of probabilities. Unlike the Markov chain model of Chapter 1, the Leslie matrix model is therefore a deterministic one.] How should we introduce environmental and/or demographic stochasticity into a Leslie model? And what can we expect from such a model? It is certainly possible to use a Markov chain approach, but since now the system states are vector and not scalar valued (that is, (N0t N1t N t) and not simply Nt) the transition matrix is more difficult to determine, and the analysis much more complex. Indeed, questions must be asked as to what assumptions are being made for example, is ceiling-type carrying capacity K the total number of individuals, or should we omit juveniles? In addition, if probabilistic the birth and death rules lead to a population size greater than K, individuals of which ages should be removed to reduce the population to size K? Rather than sampling at random from underlying probability distributions to determine the values of L in a given time step, Cohen used Markov assumptions for fluctuations in the matrix elements of L. That is, he assumed that the values of L for a given time step were influenced by their values in the previous step. Under these conditions, the geometric mean growth rate of the population is equal to the Perron root, but of course the stochastic effects add a variance factor that leads to questions such as evaluating the quasi-extinction risk. A more common approach has been to use simulation to analyze such models. In a simple experiment, Boyce showed that with values of L sampled at random the value of the geometric mean growth rate declined as the variance of the random variables was increased. For a model with age-structure, density dependence and stochasticty, simulation is surely the best approach.
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N. Arizona >> MAT >> 543 (Fall, 2008)
5.7 Stochastic models Our treatment of age-structured models thus far has been deterministic. An alternative approach to density dependence is to introduce stochastic effects to regulate population size. [It is important to note that although the px ...
N. Arizona >> FOR >> 542 (Fall, 2008)
5.7 Stochastic models Our treatment of age-structured models thus far has been deterministic. An alternative approach to density dependence is to introduce stochastic effects to regulate population size. [It is important to note that although the px ...
N. Arizona >> MAT >> 542 (Fall, 2008)
5.5 Sensitivity and elasticity As we have often noted, the exact values of the parameters are not known. We would like to believe that any results obtained would remain close for small changes in parameter values. This notion is referred to as sensit...
N. Arizona >> MAT >> 543 (Fall, 2008)
5.5 Sensitivity and elasticity As we have often noted, the exact values of the parameters are not known. We would like to believe that any results obtained would remain close for small changes in parameter values. This notion is referred to as sensit...
N. Arizona >> FOR >> 542 (Fall, 2008)
5.5 Sensitivity and elasticity As we have often noted, the exact values of the parameters are not known. We would like to believe that any results obtained would remain close for small changes in parameter values. This notion is referred to as sensit...
N. Arizona >> MAT >> 542 (Fall, 2008)
MAT 542 Spring 2006 Second Assignment Due Tuesday February 7th Density dependent models differential equations 1. In the equation 1 dN = f ( N ) in Section 3.1, we require f (0) = r, f (K) = 0, N dt f decreasing and positive on (0,K) and negati...
N. Arizona >> MAT >> 543 (Fall, 2008)
MAT 542 Spring 2006 Second Assignment Due Tuesday February 7th Density dependent models differential equations 1. In the equation 1 dN = f ( N ) in Section 3.1, we require f (0) = r, f (K) = 0, N dt f decreasing and positive on (0,K) and negati...
N. Arizona >> FOR >> 542 (Fall, 2008)
MAT 542 Spring 2006 Second Assignment Due Tuesday February 7th Density dependent models differential equations 1. In the equation 1 dN = f ( N ) in Section 3.1, we require f (0) = r, f (K) = 0, N dt f decreasing and positive on (0,K) and negati...
N. Arizona >> MAT >> 542 (Fall, 2008)
3.2 Inverse Density Dependence - A differential equation model As well as the density dependent effects that occur close to carrying capacity, there is another type which may occur in some species when the population size is very small: Some species...
N. Arizona >> MAT >> 543 (Fall, 2008)
3.2 Inverse Density Dependence - A differential equation model As well as the density dependent effects that occur close to carrying capacity, there is another type which may occur in some species when the population size is very small: Some species...
N. Arizona >> FOR >> 542 (Fall, 2008)
3.2 Inverse Density Dependence - A differential equation model As well as the density dependent effects that occur close to carrying capacity, there is another type which may occur in some species when the population size is very small: Some species...
N. Arizona >> MAT >> 542 (Fall, 2008)
B. Headcount Models 2. Density Independent Growth 2.1 A First Model Unstructured and deterministic Population models will typically have time as the independent variable, and will have as dependent variables the size of the population at that time. ...
N. Arizona >> MAT >> 543 (Fall, 2008)
B. Headcount Models 2. Density Independent Growth 2.1 A First Model Unstructured and deterministic Population models will typically have time as the independent variable, and will have as dependent variables the size of the population at that time. ...
N. Arizona >> FOR >> 542 (Fall, 2008)
B. Headcount Models 2. Density Independent Growth 2.1 A First Model Unstructured and deterministic Population models will typically have time as the independent variable, and will have as dependent variables the size of the population at that time. ...
N. Arizona >> MAT >> 542 (Fall, 2008)
4.2 Stability The Lotka-Euler equation Although we will not study the Lotka-MacKendrick-von Foerster model in full detail, we shall nevertheless use some parts of the model to obtain some results. We begin with the observation that once the initial ...
N. Arizona >> MAT >> 543 (Fall, 2008)
4.2 Stability The Lotka-Euler equation Although we will not study the Lotka-MacKendrick-von Foerster model in full detail, we shall nevertheless use some parts of the model to obtain some results. We begin with the observation that once the initial ...
N. Arizona >> FOR >> 542 (Fall, 2008)
4.2 Stability The Lotka-Euler equation Although we will not study the Lotka-MacKendrick-von Foerster model in full detail, we shall nevertheless use some parts of the model to obtain some results. We begin with the observation that once the initial ...
N. Arizona >> MAT >> 542 (Fall, 2008)
5.3 More general transition matrices So far in this chapter we have worked with a set of assumptions that say that females have a maximum age , and that there is a constant survival rate p x and a constant fecundity rate m x for each age x = 0, 1, 2 ...
N. Arizona >> MAT >> 543 (Fall, 2008)
5.3 More general transition matrices So far in this chapter we have worked with a set of assumptions that say that females have a maximum age , and that there is a constant survival rate p x and a constant fecundity rate m x for each age x = 0, 1, 2 ...
N. Arizona >> FOR >> 542 (Fall, 2008)
5.3 More general transition matrices So far in this chapter we have worked with a set of assumptions that say that females have a maximum age , and that there is a constant survival rate p x and a constant fecundity rate m x for each age x = 0, 1, 2 ...
N. Arizona >> MAT >> 542 (Fall, 2008)
1.2 Probability The probability of an event E represents how likely the event is to occur. In practice, the sum and product rules for probabilities are needed to compute theoretical probabilities. Consider the problem of finding the probability of ge...
N. Arizona >> MAT >> 543 (Fall, 2008)
1.2 Probability The probability of an event E represents how likely the event is to occur. In practice, the sum and product rules for probabilities are needed to compute theoretical probabilities. Consider the problem of finding the probability of ge...
N. Arizona >> FOR >> 542 (Fall, 2008)
1.2 Probability The probability of an event E represents how likely the event is to occur. In practice, the sum and product rules for probabilities are needed to compute theoretical probabilities. Consider the problem of finding the probability of ge...
N. Arizona >> MAT >> 542 (Fall, 2008)
MAT 543 Spring 2006 Fifth Lab: Predator-Prey Lab Objective to get a deeper understanding of Messiers paper, and to determine how critical (sensitive) some of his personal choices are to the overall conclusion. This lab requires some nonlinear regr...
N. Arizona >> MAT >> 543 (Fall, 2008)
MAT 543 Spring 2006 Fifth Lab: Predator-Prey Lab Objective to get a deeper understanding of Messiers paper, and to determine how critical (sensitive) some of his personal choices are to the overall conclusion. This lab requires some nonlinear regr...
N. Arizona >> FOR >> 542 (Fall, 2008)
MAT 543 Spring 2006 Fifth Lab: Predator-Prey Lab Objective to get a deeper understanding of Messiers paper, and to determine how critical (sensitive) some of his personal choices are to the overall conclusion. This lab requires some nonlinear regr...
N. Arizona >> MAT >> 542 (Fall, 2008)
MAT 543 Third Lab: Introductory matrix lab The objectives of this lab are (i) to reinforce notions of matrix difference equations, and (ii) to assess the maximum-age and related assumptions in Leslie matrix models. 1. Matrix Tutorial To enter a mat...
N. Arizona >> MAT >> 543 (Fall, 2008)
MAT 543 Third Lab: Introductory matrix lab The objectives of this lab are (i) to reinforce notions of matrix difference equations, and (ii) to assess the maximum-age and related assumptions in Leslie matrix models. 1. Matrix Tutorial To enter a mat...
N. Arizona >> FOR >> 542 (Fall, 2008)
MAT 543 Third Lab: Introductory matrix lab The objectives of this lab are (i) to reinforce notions of matrix difference equations, and (ii) to assess the maximum-age and related assumptions in Leslie matrix models. 1. Matrix Tutorial To enter a mat...
N. Arizona >> MAT >> 542 (Fall, 2008)
MAT 543 Fourth Lab: Model Building The objectives of this lab are to build a spreadsheet model to analyze a real world scenario, and to see if the same conclusions are drawn as in a paper in the literature. Scenario The 1960s were an unusually dry ...
N. Arizona >> MAT >> 543 (Fall, 2008)
MAT 543 Fourth Lab: Model Building The objectives of this lab are to build a spreadsheet model to analyze a real world scenario, and to see if the same conclusions are drawn as in a paper in the literature. Scenario The 1960s were an unusually dry ...
N. Arizona >> FOR >> 542 (Fall, 2008)
MAT 543 Fourth Lab: Model Building The objectives of this lab are to build a spreadsheet model to analyze a real world scenario, and to see if the same conclusions are drawn as in a paper in the literature. Scenario The 1960s were an unusually dry ...
N. Arizona >> MAT >> 542 (Fall, 2008)
6.2 The Classical Lotka-Volterra Models In this section we shall introduce explicit two-species models by way of the classical examples of Lotka and Volterra from the early 1920s. In later sections we shall replace these by more realistic models of t...
N. Arizona >> MAT >> 543 (Fall, 2008)
6.2 The Classical Lotka-Volterra Models In this section we shall introduce explicit two-species models by way of the classical examples of Lotka and Volterra from the early 1920s. In later sections we shall replace these by more realistic models of t...
N. Arizona >> FOR >> 542 (Fall, 2008)
6.2 The Classical Lotka-Volterra Models In this section we shall introduce explicit two-species models by way of the classical examples of Lotka and Volterra from the early 1920s. In later sections we shall replace these by more realistic models of t...
N. Arizona >> MAT >> 542 (Fall, 2008)
MAT 543 Second Lab: Nonlinear Dynamics Lab The objectives of this lab are (i) to reinforce notions of nonlinear iterated maps, (ii) to think about this in terms of real species and (iii) to assess delay and stochastic effects on such models. 1. The...
N. Arizona >> MAT >> 543 (Fall, 2008)
MAT 543 Second Lab: Nonlinear Dynamics Lab The objectives of this lab are (i) to reinforce notions of nonlinear iterated maps, (ii) to think about this in terms of real species and (iii) to assess delay and stochastic effects on such models. 1. The...
N. Arizona >> FOR >> 542 (Fall, 2008)
MAT 543 Second Lab: Nonlinear Dynamics Lab The objectives of this lab are (i) to reinforce notions of nonlinear iterated maps, (ii) to think about this in terms of real species and (iii) to assess delay and stochastic effects on such models. 1. The...
N. Arizona >> MAT >> 542 (Fall, 2008)
3.8 Stochastic Effects To complete this chapter, we should now return to the topics that formed much of the discussion in Chapter 1, namely the effects of demographic and environmental stochasticity. It is clear that density-dependence must increase ...
N. Arizona >> MAT >> 543 (Fall, 2008)
3.8 Stochastic Effects To complete this chapter, we should now return to the topics that formed much of the discussion in Chapter 1, namely the effects of demographic and environmental stochasticity. It is clear that density-dependence must increase ...
N. Arizona >> FOR >> 542 (Fall, 2008)
3.8 Stochastic Effects To complete this chapter, we should now return to the topics that formed much of the discussion in Chapter 1, namely the effects of demographic and environmental stochasticity. It is clear that density-dependence must increase ...
N. Arizona >> MAT >> 542 (Fall, 2008)
A. Background 1. Background 1.1 The Modeling Process Mathematical models are usually built following the scheme indicated below. A. Identify the problem of interest and the objective of the model that is to be built. B. Start simply. Make a set of as...
N. Arizona >> MAT >> 543 (Fall, 2008)
A. Background 1. Background 1.1 The Modeling Process Mathematical models are usually built following the scheme indicated below. A. Identify the problem of interest and the objective of the model that is to be built. B. Start simply. Make a set of as...
N. Arizona >> FOR >> 542 (Fall, 2008)
A. Background 1. Background 1.1 The Modeling Process Mathematical models are usually built following the scheme indicated below. A. Identify the problem of interest and the objective of the model that is to be built. B. Start simply. Make a set of as...
N. Arizona >> MAT >> 542 (Fall, 2008)
5. Matrix Difference Equations 5.1 Leslie Matrix Models In this section, we shall continue on from Section 4.5 to develop a discrete time model of age-structured population dynamics that takes the form of a matrix difference equation. Recall that the...
N. Arizona >> MAT >> 543 (Fall, 2008)
5. Matrix Difference Equations 5.1 Leslie Matrix Models In this section, we shall continue on from Section 4.5 to develop a discrete time model of age-structured population dynamics that takes the form of a matrix difference equation. Recall that the...
N. Arizona >> FOR >> 542 (Fall, 2008)
5. Matrix Difference Equations 5.1 Leslie Matrix Models In this section, we shall continue on from Section 4.5 to develop a discrete time model of age-structured population dynamics that takes the form of a matrix difference equation. Recall that the...
N. Arizona >> MAT >> 542 (Fall, 2008)
MAT 543 Spring 2006 First Lab: Growth Rates The objectives of this lab are (1) to gain an intuitive feel for the parameters r and , to appreciate that wildlife populations have small growth rates and to appreciate how hard it is to estimate these p...
N. Arizona >> MAT >> 543 (Fall, 2008)
MAT 543 Spring 2006 First Lab: Growth Rates The objectives of this lab are (1) to gain an intuitive feel for the parameters r and , to appreciate that wildlife populations have small growth rates and to appreciate how hard it is to estimate these p...
N. Arizona >> FOR >> 542 (Fall, 2008)
MAT 543 Spring 2006 First Lab: Growth Rates The objectives of this lab are (1) to gain an intuitive feel for the parameters r and , to appreciate that wildlife populations have small growth rates and to appreciate how hard it is to estimate these p...
N. Arizona >> MAT >> 542 (Fall, 2008)
NORTHERN ARIZONA UNIVERSITY DEPARTMENT OF MATHEMATICS AND STATISTICS SYLLABUS AND COURSE INFORMATION Spring 2006 MAT 542 and 543 INSTRUCTOR INFORMATION Instructor: Dr. Terence Blows Office: AMB 178 Phone: 523-6863 Hours: 11:00-12:30 TTh E-mail: Teren...
N. Arizona >> MAT >> 543 (Fall, 2008)
NORTHERN ARIZONA UNIVERSITY DEPARTMENT OF MATHEMATICS AND STATISTICS SYLLABUS AND COURSE INFORMATION Spring 2006 MAT 542 and 543 INSTRUCTOR INFORMATION Instructor: Dr. Terence Blows Office: AMB 178 Phone: 523-6863 Hours: 11:00-12:30 TTh E-mail: Teren...
N. Arizona >> FOR >> 542 (Fall, 2008)
NORTHERN ARIZONA UNIVERSITY DEPARTMENT OF MATHEMATICS AND STATISTICS SYLLABUS AND COURSE INFORMATION Spring 2006 MAT 542 and 543 INSTRUCTOR INFORMATION Instructor: Dr. Terence Blows Office: AMB 178 Phone: 523-6863 Hours: 11:00-12:30 TTh E-mail: Teren...
N. Arizona >> MAT >> 542 (Fall, 2008)
3.6 Time Lag Models The logistic differential equation is one in which the density-dependent effects are considered to act immediately. The rate of change of N(t) at time t depends on the population size at the same time t. It is reasonable to ask ho...
N. Arizona >> MAT >> 543 (Fall, 2008)
3.6 Time Lag Models The logistic differential equation is one in which the density-dependent effects are considered to act immediately. The rate of change of N(t) at time t depends on the population size at the same time t. It is reasonable to ask ho...
N. Arizona >> FOR >> 542 (Fall, 2008)
3.6 Time Lag Models The logistic differential equation is one in which the density-dependent effects are considered to act immediately. The rate of change of N(t) at time t depends on the population size at the same time t. It is reasonable to ask ho...
N. Arizona >> MAT >> 542 (Fall, 2008)
F P B1 B2 B3 m1 m2 Immature 30 16.6 20.32 16.92 16.65 15.18 14.32 13.3 12.44 11.59 10.83 10.1 9.43 8.8 8.21 7.66 7.15 6.67 6.23 5.81 5.42 5.06 4.72 4.41 4.12 3.84 3.58 3.34 3.12 2.91 2.72 2.54 2.37 2.21 2.06 1.92 1.8 1.68 1.56 1.46 1.36 1.27 1.19 0....
N. Arizona >> MAT >> 543 (Fall, 2008)
F P B1 B2 B3 m1 m2 Immature 30 16.6 20.32 16.92 16.65 15.18 14.32 13.3 12.44 11.59 10.83 10.1 9.43 8.8 8.21 7.66 7.15 6.67 6.23 5.81 5.42 5.06 4.72 4.41 4.12 3.84 3.58 3.34 3.12 2.91 2.72 2.54 2.37 2.21 2.06 1.92 1.8 1.68 1.56 1.46 1.36 1.27 1.19 0....
N. Arizona >> FOR >> 542 (Fall, 2008)
F P B1 B2 B3 m1 m2 Immature 30 16.6 20.32 16.92 16.65 15.18 14.32 13.3 12.44 11.59 10.83 10.1 9.43 8.8 8.21 7.66 7.15 6.67 6.23 5.81 5.42 5.06 4.72 4.41 4.12 3.84 3.58 3.34 3.12 2.91 2.72 2.54 2.37 2.21 2.06 1.92 1.8 1.68 1.56 1.46 1.36 1.27 1.19 0....
N. Arizona >> MAT >> 542 (Fall, 2008)
MAT 542 Spring 2006 Phase Plane Fifth Homework Due Tuesday April 11th 1. Set up a Lotka-Volterra type-model for (a) commensalism where one species gains from another, but the second gains no advantage/disadvantage from the first, and (b) amensali...
N. Arizona >> MAT >> 543 (Fall, 2008)
MAT 542 Spring 2006 Phase Plane Fifth Homework Due Tuesday April 11th 1. Set up a Lotka-Volterra type-model for (a) commensalism where one species gains from another, but the second gains no advantage/disadvantage from the first, and (b) amensali...
N. Arizona >> FOR >> 542 (Fall, 2008)
MAT 542 Spring 2006 Phase Plane Fifth Homework Due Tuesday April 11th 1. Set up a Lotka-Volterra type-model for (a) commensalism where one species gains from another, but the second gains no advantage/disadvantage from the first, and (b) amensali...
N. Arizona >> MAT >> 542 (Fall, 2008)
MAT 542 Spring 2004 Fourth Assignment Due Thursday March 4th Post-breeding and pre-breeding models 1. a) Write the following post-breeding generalized Leslie matrix for Red-Cockaded Woodpeckers (Maguire, Wilhere and Dong Population viability anal...
N. Arizona >> MAT >> 543 (Fall, 2008)
MAT 542 Spring 2004 Fourth Assignment Due Thursday March 4th Post-breeding and pre-breeding models 1. a) Write the following post-breeding generalized Leslie matrix for Red-Cockaded Woodpeckers (Maguire, Wilhere and Dong Population viability anal...
N. Arizona >> FOR >> 542 (Fall, 2008)
MAT 542 Spring 2004 Fourth Assignment Due Thursday March 4th Post-breeding and pre-breeding models 1. a) Write the following post-breeding generalized Leslie matrix for Red-Cockaded Woodpeckers (Maguire, Wilhere and Dong Population viability anal...
N. Arizona >> MAT >> 542 (Fall, 2008)
2.2 Linear Deterministic Models The simplest choice of model is one that is density independent, that is, it uses the assumption that birth and death rates are assumed to be independent of population size. This leads to a linear model, which has an a...
N. Arizona >> MAT >> 543 (Fall, 2008)
2.2 Linear Deterministic Models The simplest choice of model is one that is density independent, that is, it uses the assumption that birth and death rates are assumed to be independent of population size. This leads to a linear model, which has an a...
N. Arizona >> FOR >> 542 (Fall, 2008)
2.2 Linear Deterministic Models The simplest choice of model is one that is density independent, that is, it uses the assumption that birth and death rates are assumed to be independent of population size. This leads to a linear model, which has an a...
N. Arizona >> MAT >> 542 (Fall, 2008)
5.6 Density dependence In this section we introduce density dependent effects to limit a population to a carrying capacity. It is difficult to know exactly how to incorporate density dependence into an age-structured model. A number of questions come...
N. Arizona >> MAT >> 543 (Fall, 2008)
5.6 Density dependence In this section we introduce density dependent effects to limit a population to a carrying capacity. It is difficult to know exactly how to incorporate density dependence into an age-structured model. A number of questions come...
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