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## Über dieses Buch

This monograph has been heavily influenced by two books. One is Ren­ shaw's [82] work on modeling biological populations in space and time. It was published as we were busily engaged in modeling African bee dispersal, and provided strong affirmation for the stochastic basis for our ecological modeling efforts. The other is the third edition of Jacquez' [28] classic book on compartmental analysis. He reviews stochastic compartmental analysis and utilizes generating functions in this edition to derive many useful re­ sults. We interpreted Jacquez' use of generating functions as a message that the day had come for modeling practioners to consider using this powerful approach as a model-building tool. We were inspired by the idea of using generating functions and related methods for two purposes. The first is to integrate seamlessly our previous research centering in stochastic com­ partmental modeling with our more recent research focusing on stochastic population modeling. The second, related purpose is to present some key research results of practical application in a natural, user-friendly way to the large user communities of compartmental and biological population modelers. One general goal of this monograph is to make a case for the practical utility of the various stochastic population models. In accordance with this objective, we have chosen to illustrate the various stochastic models, using four primary applications described in Chapter 2. In so doing, this mono­ graph is based largely on our own published work.

## Inhaltsverzeichnis

### 1. Overview of Models

Abstract
Many types of mathematical models have been proposed in the literature for describing biological populations [6, 19, 28, 69, 80, 82]. One might argue that the wide variety of such models is a natural consequence of the great diversity of the overall objectives for such models. We start by discussing briefly the general modeling objectives upon which this monograph is based.
James H. Matis, Thomas R. Kiffe

### 2. Some Applications

Abstract
Each of the subsequent chapters contains one or more simple examples to illustrate the concepts developed in the chapter. In addition to these simple examples, four applications which we have published in the recent literature will be used multiple times, in two or more chapters. Two or more stochastic models will be proposed to describe each of these four data sets, and a comparative analysis will help to clarify the relative advantages and disadvantages of the various models. A description of these four key applications follows.
James H. Matis, Thomas R. Kiffe

### 3. Basic Methodology for Single Population Stochastic Models

Abstract
This chapter introduces the basic theoretical tools utilized subsequently for solving single population stochastic models. Most of these tools are developed more extensively in the basic textbooks in stochastic processes, including the leading applied texts by Bailey [4], Chiang [12] and Renshaw [82]. The basic methodology is also described and illustrated specifically as applied to standard compartmental models in Jacquez [28]. This monograph will build on Jacquez’ development, most notably by including “births” into the stochastic compartmental models.
James H. Matis, Thomas R. Kiffe

### 4. Linear Immigration-Death Models

Abstract
Consider first modeling a single population of sizeX(t)with the linear death rate
$$\mu x = aX$$
(4.1)
and immigration rate I. This very simple case of the general rates in (3.40) extends the illustration in Chapter 3 by relaxing the assumption of an initial population size zero. The model, as previously noted, is called the linear immigration-death (LID) model, and it is widely used as the standard stochastic one-compartmental model. This model is developed thoroughly in [28] in a compartmental context. A simple application developed subsequently is modeling the bioaccumulation of mercury, described in Section 2.4. We have also used the model extensively to describe the passage of digesta in ruminant animals, see e.g. [15, 16, 59].
James H. Matis, Thomas R. Kiffe

### 5. Linear Birth-Immigration-Death Models

Abstract
Consider now modeling the single population of sizeX (t)with linear birth and death rates:
$${\lambda _X} = {a_1}Xand\mu x = {a_1}X.$$
(5.1)
The immigration rate is again assumed to beI.This is called the LBID model. Clearly this model with linear (density-independent) growth rates could hold in the case of al> a2only for periods of initial population growth. The model will be illustrated for such initial population growth for both the African bee and the muskrat examples in Chapter 2.
James H. Matis, Thomas R. Kiffe

### 6. Nonlinear Birth-Death Models

Abstract
Consider now modeling the single population of sizeX(t)with population rates
$$\lambda _X = \left\{ \begin{gathered} a_1 X - b_1 X^{s + 1} for X < (a_1 /b_1 )^{1/s} \hfill \\ 0 otherwise \hfill \\ \end{gathered} \right.$$
(6.1)
$$\mu _X = a_2 X + b_2 X^{s + 1}$$
(6.2)
for integers >1. These are called nonlinear rates in ecological population modeling because the per capita rates, i.e.Ax/X, are obviously functions ofX, as discussed in Section 3.7. The aiare called the intrinsic rates, and are interpreted as the per-capita birth and death rate coefficients for small initial population sizes, before population density pressures are of any practical consequence. Theb i are the crowding coefficients which add density-dependency to the model. Their inclusion may in principle yield long-term, equilibrium solutions for the model.
James H. Matis, Thomas R. Kiffe

### 7. Nonlinear Birth-Immigration-Death Models

Abstract
Consider now adding immigration at rateIto the nonlinear birth-death model. This new model is more realistic biologically in many applications, including the AHB and muskrat problems, however it is largely ignored in the literature. Perhaps there are two reasons for this. One is that the deterministic models become more unwieldy whenIis included. The other reason is that inclusion of biologically reasonable levels ofIseldom have a large impact on the qualitativeshapeof the deterministic solution. Consequently, in practice a statistical analysis would often fail to show that the inclusion of the parameter leads to a substantial improvement in the goodness-of-fit of the model to the data. In short, though there are often compelling biological reasons to include immigration, it is usually excluded from the model because it complicates thedeterministicmodel and because its inclusion is typically not statistically “significant”.
James H. Matis, Thomas R. Kiffe

### 8. Standard Multiple Compartment Analysis with the Deterministic Model

Abstract
Compartmental modeling is widely used in pharmacokinetics and in physiological modeling, for which there is a vast literature. In addition to Jacquez [28], leading texts include [3, 22, 85]. As a simple example, the mercury bioaccumulation problem in Section 4.2.2 illustrates the need for a multi-compartment analysis. The techniques have also been applied to ecosystem modeling (see e.g. [52]).
James H. Matis, Thomas R. Kiffe

### 9. Basic Methodology for Multiple Population Stochastic Models

Abstract
This chapter generalizes the basic theoretical tools developed in Chapter 3 for single population stochastic models to the case of multiple population models. These tools also are considered extensively in the leading texts on applied stochastic processes, including [4, 9, 12, 82]. Jacquez [28] illustrates their use in a standard compartmental context.
James H. Matis, Thomas R. Kiffe

### 10. Linear Death-Migration Models

Abstract
Consider now modeling n populations connected by linear migration, with each population having both immigration and death events occurring at linear rates. Specifically, the assumptions for unit changes in this linear system fromt to t + Δt, using the general formulation in (9.3) and (9.4), would be:
1
Prob#x007B;X, will increase by 1 due to immigration} = I i Δt

2
Prob#x007B;Xi will decrease by 1 due to death} = μ i X i Δt

3
Prob{Xi will increase by 1 and X; will decrease by 1 due to migration} = k ij X j Δt, for ij.

James H. Matis, Thomas R. Kiffe

### 11. Linear Immigration-Death-Migration Models

Abstract
Consider now the stochastic LIDM model, which generalizes the basic single-population LID model introduced in Chapter 4. A simple case of this LIDM model is considered in Chapter 9 to illustrate the basic methodology for multiple populations. This chapter develops a more general case, specifically the case with multiple inputs and two-way migration between compartments. The corresponding deterministic model is given in (8.4) of Chapter 8, with its solution in (8.5) and (8.6), and with the illustration of mercury uptake by fish in Section 8.3.1.
James H. Matis, Thomas R. Kiffe

### 12. Linear Birth-Immigration-Death Migration Models

Abstract
Consider now adding births to the previous linear multidimensional models discussed in Chapters 10 and 11. The assumptions are then
1
Prob{Xiwill increase by 1 due to immigration{ = I i Δt

2
Prob{Xiwill decrease by 1 due to death{=µ i X i Δt

3
Prob {Xiwill increase by 1 due to birth{=λ i X i Δt

4
Prob{Xiwill increase by 1 and Xjwill decrease by 1 due to migration{= k ij X j Δt for i,j=1,2 with ij

James H. Matis, Thomas R. Kiffe

### 13. Nonlinear Birth-Death-Migration Models

Abstract
Consider generalizing the single-population models with nonlinear kinetics given in Chapters 6 and 7 to the multiple population case. Chapters 13 and 14 consider two general models, very different from one another, to illustrate the application of the methodology introduced in Chapter 9 to various multi-population models with nonlinear rates.
James H. Matis, Thomas R. Kiffe

### 14. Nonlinear Host-Parasite Models

Abstract
Modeling the dynamics of interacting species is central in ecological theory, and there is a vast literature describing deterministic and stochastic models for such interactions (see e.g. [65, 66, 69, 73, 80]). A number of classic models for competition between multiple species and for host-parasite (and predator-prey) relationships have been formulated and widely investigated. This chapter investigates just one of these models, a modified Volterra host-parasite model, to illustrate the application of the methodology outlined in Chapter 9 which focuses on the underlying cumulant structure of the stochastic model.
James H. Matis, Thomas R. Kiffe

### Backmatter

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