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2.5.1 Finding the right compromise

Because of their sensitivity to a wide range of biotic as well as abiotic parameters, a complete model description of a biological system (ecosystem, genetic network, etc) would consist of an extremely large number of parameters. This presents two problems. First, to write and parameterise accurately a complete model would be a feat in itself. Temperature, humidity, and carbon dioxide concentration are known to impact reproductive cycles, functional responses and mortality of various animals (see e.g. examples given in Chapter 7 in Cohen (1978), Gerson et al. (2003), and Weeden et al. (2007)). This inherent complexity introduces process errors which arise if an experiment is to be repeated under conditions that are not identical. It also motivates some of the criticisms against experiments in laboratory, that is under highly controlled environments (which could aid in parameterising the system), in that they do not accurately mimic field or real world conditions. These features represent the process errors.

Second, even for models of intermediate complexity, there are additional obstacles on the way: its mathematical analysis is only possible by numerical methods. Such calculations tend to be computationally intensive.

### A local preview: with or without continuous control

We may obtain some insights on the system’s behaviour if we consider two regimes: one under continuous control and one without it. We write this model as: ( x_ = f(x) g(x)y y_ = h(x)y my + (3.3)

We analyse the local behaviour of the system – setting > 0 gives the controlled local trajectories while = 0 corresponds to the uncontrolled system. At equilibrium, ( f(x) g(x)y = 0 h(x)y my + = 0 (3.4) From Hypotheses (H1)–(H3) on the system functions, one of the solutions of f(x) g(x)y = 0 is x = 0. Setting x = 0 in the second equation of (3.4), we calculate the zero-pest equilibrium (x; y).

#### Stability with affine parameterisation

In the affine case, the functions read (x) = ax and (y) = by + 1, where a; b are positive constants. This formulation corresponds to that of Buffoni et al. (2005) for density-dependent trophic responses; our analysis extends his work to the case of impulsive biological control. It is a generalisation of the Beddington-DeAngelis trophic response. The results of this section therefore encompass those obtained by Zhang and Chen (2006) and Nundloll et al. (2010b) on biological control with explicit Beddington-DeAngelis functions. The stability conditions are described by the following theorem.

**Nonlinear parameterisation**

When the parameterisation of (x) and (y) is nonlinear, only general trends in the behaviour of the release rate with respect to the biological processes can be identified. While the precise value of the minimal release rate, which we shall denote by ~(T), is unknown, we know however that it is bounded as in (4.33). Each of the bounds have the same form as for the affine case: each of the minimal release rate bounds will decrease with aig0(0) and increase with bi. The actual ~(T) will therefore generally tend to increase with respect to each aig0(0) and bi within a tube defined by these two bounds.

**Additional questions: beyond sector modelling**

The sector formulation in this chapter generate a maximal family of models that yield two types of stability conditions for the zero-pest solution: the first is a biological and necessary condition, and the second, a managerial one. This observation indicates that the result of the stability analysis is not related to the explicit form of the trophic responses (such as the Beddington-DeAngelis function, which yields these two types of conditions), but rather, to the structure of the trophic response’s parameters. In our final paper (Nundloll et al., 2010a), we presented the structural changes that occur to the stability of the zero-pest solution when the parameters of the trophic response cannot be be enclosed within sectors. We highlighted the need to discriminate between three levels of interference:‘within sectors or moderate’, ‘too weak’, and ‘too strong’. We were also able to generalise completely Buffoni et al. (2005)’s formulation for both density and ratiodependence

**Table of contents :**

Contents

List of Figures

List of Tables

Abstract

Résumé

Acknowledgements

**1 Introduction **

1.1 Understanding biological control

1.1.1 Who kills who?

1.1.2 Kill the pest!

1.1.3 Use predators

1.1.4 Augmentative control

1.1.5 Predator types

1.1.6 Integrated Pest Management (IPM)

1.1.7 Some vital statistics

1.2 Scope of this thesis

1.3 Detailed plan

**2 Making a choice: modelling approach **

2.1 Building the model

2.1.1 Keeping count

2.1.2 Some key assumptions

2.1.3 Hybrid dynamics

2.1.4 Canonical form

2.2 Two mathematical properties of the impulsive model

2.3 Why go impulsive?

2.4 Why go qualitative? – I. Structural advantages

2.4.1 Predator-prey processes

2.4.2 Asymptotic behaviour

2.5 Why go qualitative? – II. Field realities

2.5.1 Finding the right compromise

2.5.2 One model to rule them all

2.6 Summary of model characteristics and aims

2.6.1 Modelling features

2.6.2 Analysis

**3 Groundwork: a simple model and analytic methods **

3.1 A simple model

3.2 Mathematical analysis of the simple model

3.2.1 A local preview: with or without continuous control

3.2.2 Positivity of the system and existence of the zero-pest solution .

3.2.3 Stability of the zero-pest solution

3.2.4 A note on the computation of the global attractivity condition

3.2.5 Interpreting the LAS and GAS conditions

3.3 Pest control strategy

3.3.1 The minimal predator release rate

3.3.2 The period of release

**4 Predators interfering with each other during predation **

4.1 The premises

4.2 The model

4.3 The zero-pest solution

4.3.1 Stability with affine parameterisation

4.3.2 Stability with nonlinear parameterisation

4.4 The minimal predator release rate

4.4.1 Affine parameterisation

4.4.2 Nonlinear parameterisation

4.5 The pest evolution rate

4.5.1 Affine parameterisation

4.5.2 Nonlinear parameterisation

4.6 The pest control strategy

4.6.1 Choice of the predator species

4.6.2 The predator releases

4.6.3 Additional questions: beyond sector modelling

**5 Cannibalism among predators **

5.1 Understanding cannibalism

5.1.1 Cannibalism in the biomathematical jargon

5.1.2 Impact of cannibalism on augmentative pest control

5.1.3 General model of cannibalism

5.2 Mechanism I: Territoriality

5.2.1 The premises

5.2.2 The model and specific hypotheses

5.2.3 Existence of the zero-pest solution

5.2.4 Stability of the zero-pest solution

5.2.5 The minimal release rate

5.3 Mechanism II: Hunger and fighting for survival

5.3.1 The premises

5.3.2 The model and specific hypotheses

5.3.3 Existence of the zero-pest solution

5.3.4 Stability of the zero-pest solution

5.4 Mechanism III: Diet, alternate prey & intraguild predation

5.4.1 The premises

5.4.2 The model and specific hypotheses

5.4.3 Existence and stability of the zero-pest solution

5.4.4 The Kohlmeier- Ebenhöh model

5.5 The pest control strategy

5.5.1 Predator releases

5.5.2 The selection of the predator species

**6 Partial harvest effects **

6.1 The premises

6.2 The model

6.3 Existence of the zero-pest solution

6.3.1 Releases more frequent than harvests

6.3.2 Releases less frequent than harvests

6.4 Stability of the pest-free solution

6.4.1 The methodology

6.4.2 Releases more frequent than harvests

6.4.3 Releases less frequent than harvests

6.5 The minimal release rate

6.5.1 Dependence in biological processes

6.5.2 Dependence in the period of release

6.5.3 Dependence in partial harvesting parameters

6.6 The pest control strategy

**7 Practical guidelines and experimental results **

7.1 Guidelines

7.1.1 Matching data with the model

7.1.2 Simple augmentative control program

7.1.3 Interfering predators

7.1.4 Overcrowding

7.1.5 Integrated Pest Management (or partial harvests)

7.2 Experiments with the predator Neoseiulus cucumeris to control the pestm Frankliniella occidentalis

7.2.1 Experiment 1: intrapredator interference

7.2.2 Experiment 2: release frequencies and pest control

**8 Conclusion and prospects **

8.1 Summary

8.2 Discussion

**A Why reduce pesticide usage **

**Bibliography **