Population and Community Ecology for Insect Management and Conservation

Johann Baumgartner, Pietro Brandmayr, Bryan F.J. Manly

• Publisher: CRC Press
• Year: 2020
• eBook ISBN: 9781000673654
• Topic: Biological Sciences
• Subtopic: Entomology
• Index: Biological Sciences

One of the themes of the 20th International Congress of Entomology held in Florence in August 1996 was Ecology and Population Dynamics, with papers presented on single species dynamics, population interactions, and community ecology. This book contains a selection of the papers that were presented, and gives a late-1990s picture of the latest research in this fast developing area.

1 Single species dynamics

Sampling and modelling of insect populations

Bryan F. J. Manly

University of Otago, Dunedin, New Zealand

ABSTRACT: This paper gives a general review of various topics that are discussed in some of the papers that follow, under the general theme of the sampling and modelling of insect populations. The topics covered includes models for the development of age-structured or stage-structured populations, various questions related to studying the abundance of a population or several populations over a number of generations, the dispersion of a population in space, and the measurement of the correlation of spatial patterns, and testing this for significance. An attempt is made to put recent developments into an historic framework.

1 General Themes

There are four general themes that are discussed by the Invited Speakers for the Symposium on Sampling and Modelling:

• models for the development of age-structured or stage-structured populations (Di Cola and Baumgärtner, Worner);
• sampling and modelling questions related to studying the abundance of a population or several populations over a number of generations (McArdle);
• modelling of the dispersion of a population in space (Sharov); and
• the measurement of the correlation of spatial patterns, and testing this for significance (Liebhold and Sharov, Perry).

Alternatively, these can be thought of as two topics concerning populations in time, and two topics concerning populations in space. All the contributions are characterised as having a quantitative emphasis, and being quite mathematical in their approaches.

In this introductory paper I have chosen to look at the themes one by one, with something of an historical perspective, and to fit the papers as best I can within the framework that this produces. Several of these topics are covered in more detail in a longer review that I have produced recently (Manly, 1994a).

2 Modelling Age-Structured and Stage-Structured Populations

There are three different analytical approaches that are used to model reproducing populations. The first of these approaches is based on the integral equation model of Sharpe and Lotka (1911) and Lotka (1939) which relates the number of births at time t to the number of individuals born at earlier times, the survival rates to different ages, and the reproduction rates at different ages. The second approach uses continuous time differential equations for either age-structured or stage-structured populations, as exemplified for example by the work of Metz et al. (1988) and Wood and Nisbet (1991), building on the equations of McKendrick (1926) and von Foerster (1959). The third approach uses discrete time matrix models for age-structured or stage-structured populations, as described for example by Caswell (1989) and Manly (1990, Chapter 6), building on the work of Bernardelli (1941), Lewis (1942), and Leslie (1945, 1948).

As noted by Caswell (1989, p. 24) various arguments have been used in favour of using one of these approaches for modelling instead of the others but the choice is really a matter of personal taste. However, matrix models are relatively easy to adapt to changed assumptions, are well suited for numerical calculations, and they reflect the true nature of many insect life cycles with discrete stages. They will therefore often be the most appropriate of the classical analytical methods in entomological applications.

The basis of the Bernadelli-Lewis-Leslie model and its extension for stage-structured populations is as follows. Let n(x, t) denote the number of females in the age group x at time t, p(x) denote the probability that a female in the age group x at time t will survive to be in the age group x + 1 at time t + 1, and B(x) denote the average number of female offspring born to females aged from x to x + 1 in a unit period of time that survive to the end of that period. The number of females in age group x at time t + 1 will then be the sum of the offspring from females of different ages, so that

$n(0,t+1)=B\left(0\right)n(0,t)+B\left(1\right)n(1,t)+\dots +B\left(k\right)n(k,t),$

where k + 1 is the maximum possible age. It also follows from the definitions that

$n(x+1,t+1)=p\left(x\right)n(x,t),$

for x = 0,1,..., k – 1. These equations can be written together as the matrix equation

$\left[\begin{array}{c}\text{n}(0,\text{t}+1)\\ \text{n}(1,\text{t}+1)\\ ⋮\\ \text{n}(\text{k},\text{t}+1)\end{array}\right]=\left[\begin{array}{ccccc}\text{B}\left(0\right)& \text{B}\left(1\right)& \dots & \text{B}(\text{k}-1)& \text{B}\left(\text{k}\right)\\ \text{p}\left(0\right)& 0& \dots & & 0\\ ⋮& & & & \\ 0& 0& \dots & \text{p}(\text{k}-1)& 0\end{array}\right]\left[\begin{array}{c}\text{n}(0,\text{t})\\ \text{n}(1,\text{t})\\ ⋮\\ \text{n}(\text{k},\text{t})\end{array}\right]$

or

${\mathit{N}}_{t+1}={\mathit{\text{MN}}}_t\mathrm{.}$

It then follows that

${\mathit{N}}_t={\mathit{M}}^t{\mathit{N}}_0\mathrm{.}(1)$

The matrix M, whose elements are the fecundity rates B(x) and the survival probabilities p(x), is often called the Leslie matrix. The last equation shows that the numbers in different age groups at an arbitrary time t are determined by the numbers in the age groups at time zero (N₀) and the Leslie matrix raised to the power t. Subject to certain mild assumptions it is possible to show that a population following this model will eventually reach a stable distribution for the relative numbers of individuals with different ages, and be growing or declining at a constant rate, with the long term behaviour of the population determined by the dominant eigenvalue of the Leslie matrix.

Lefkovitch (1963, 1964a, 1964b, 1965) modified the Bernardelli-Leslie-Lewis model to allow a population to be grouped by life stages rather than by age by allowing the number in stage j at time t + 1 to depend on the numbers in all previous stages at time t. Thus if f_j(t) is the number of individuals in stage j at time t then for q stages his model is, in matrix notation,

$\left[\begin{array}{c}{f}_1(t+1)\\ f_2(t+1)\\ ⋮\\ f_q(t+1)\end{array}\right]=\left[\begin{array}{cccc}{m}_{11}& m_{12}& \dots & m_{1q}\\ m_{21}& m_{22}& & m_{2q}\\ & & & \\ m_{q1}& m_{q2}& \dots & m_{\mathit{\text{qq}}}\end{array}\right]\left[\begin{array}{c}{f}_1\left(t\right)\\ f_2\left(t\right)\\ ⋮\\ f_q\left(t\right)\end{array}\right]$

or

${\mathit{F}}_{t+1}={\mathit{\text{MF}}}_t,$

so that

${\mathit{F}}_t={\mathit{M}}^t{\mathit{F}}_0\mathrm{.}(2)$

The typical entry in the matrix M in equation (2), m_ij, reflects how the number in stage i at time t + 1 depends on the number in stage j at time t. Equation (2) is similar to equation (1) but the matrix M of the latter equation does not have the simple structure of a Leslie matrix, with its many necessarily zero elements. There is an implicit assumption with Lefkovitch’s model that the age distribution within stages is constant enough to make any variation in the m_ij values with time unimportant...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Single species dynamics
• 2 Population interactions
• 3 Community ecology
• Author index