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Why go qualitative? – II. Field realities
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 reproduc-tive 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 parameteris-ing 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.
Observation errors also contribute to the difficulty in modelling the biological processes in a predator-prey system.
In many cases, this is due to the problem of scale both with respect to the observer (the human being) and with respect to the observed patch. For instance, tracking insects that may barely be a fraction of millimetres wide over hectares of crops cannot be done with the same precision as counting larger animals (of size of the same order or larger than humans). Likewise, measuring fish stock accurately is impossible as the oceans span the planet without borders: here it is the size of the patch which is extremely big. The problem of scale also encompasses the idea of refuges where the living organisms hide and remain out of sight of the observer’s eye. This makes the process of keeping count extremely tricky.
Sensitivity to the observer is also a source of measurement error. An observer can rep-resent an enemy threat or physically hurt individuals of the species under obervation when manipulating them. Small organisms can be hurt or killed when manipulated roughly. This is the case with Frankliniella occidentalis (Pergande) and Amblyseiulus cucumeris (Oudemans) – which are a major plant pest and one of its biological control agents and only a fraction of millimetres in length – that are considered in our experiments reported in Chapter 7 (Vitte, 2008). They can also show defensive behaviour provoked by the presence of the observer who represents a potential enemy threat. Larson (2003) captures the issue best (see Figure 2.5). This is another source of observation error.
Cartoon removed for copyright reasons
Figure 2.5: Human observation can be disruptive and yield inaccurate results on the behaviour of living organisms. Larson (2003) provides a humourous take on the be-haviour of cows in the absence of humans.
Finding the right compromise between complexity and simplicity is one of the aims of math-ematical modelling (Bernard, 2004)3. Tackling process and observation errors is, however, no mean feat.
Suppose, as biomathematicians, we decide to focus on a set of processes only and describe them in detail. Here there is still a risk that the analysis yields results that are related to modelling artefacts (Clark, 1990). Take for example the Holling Type II functional response (Holling, 1961). The general shape of a data cloud can also be fitted by Ivlev (1961)’s function which is instead based on satiation of predators. Emlen (1973) (in Real (1977)) offers an alternative food handling explanation, where the majority of time is spent feeding on either prey to reach a similar form. In fact, there are also probably other functions that can fit a noisy data set representing the process at each population size4. Inevitably the use of either function (here, Holling’s or Emlen’s) or either interpretation of parameters (here, Holling’s or Emlen’s) will have consequences on the final interpretation of the analytical results from the model.
This problem also arises when fitting models directly to time-series data on populations. Because of noise (observation or process errors) in the collected data, models that are struc-turally different can be fitted – after adequate parameter adjustments – to the same time-series. Jost and Arditi (2000) (see Figure 2.6) show how data artificially generated with a prey-only dependent functional response can also be fitted by a model with a ratio-dependent (that is prey and predator) functional response, and vice-versa5. This indicates a fundamen-tal difficulty in identifying mechanisms underlying the functional processes that are used to build a model and interpret the results of its analysis. The model with prey-dependent response can fit data generated by the ratio-dependent model.
Remark: In order to tackle large models analytically, a biomathematician can also resort to a variety of model reduction techniques (see e.g. Beck et al. (1996)). These are not always very straightforward to implement. Other approaches exist: Cohen (1978)’s study of niche spaces resorts to graphical methods to reduce niche space and food web sizes. As he points out, this approach does not take into account the dynamical processes between the individual species of the web.
4Examples of process data sets are given in Begon et al. (1996). We also illustrate the applications of our results in Chapter 7 with such data sets One way to reduce the risks of modelling artefacts in a simple model is to focus on the general trends that a process follows with respect to the state variables. Gouzé (1996) for instance remarked that biological systems were characterised by some general properties: positive variables, dynamic state changes being dependent on the sum or difference of some positive processes etc. Such a modelling approach is said to be qualitative or generic; the resultant model is referred to as a meta-model(D’Onofrio, 2008).
By focusing on general trends, the meta-model smoothes out uncertainties or errors in parameterising various process functions. As discussed previously in Section 2.4.2 it also yields results that apply to a wide class of models instead of being specific to one particular explicitly formulated model. Once we have identified the stability conditions for the solution of a generic model, the result for a specific model is recovered simply by plugging the explicit function’s parameters.
Lakmeche and Arino (2000)’s analysis of a generic model inspired by chemotherapeutic treatment models gives generic results that are applicable to the predator-prey models with impulsive dynamics (e.g. in this thesis, Negi and Gakkhar (2007)). The use of generic func-tions is common in chemostat modelling: the substrate consumption is often modelled as some function of the substrate satisfying some basic properties (see e.g. Bernard (2004)).
Among predator-prey models, qualitative modelling is rare. Nonetheless, there are com-mon attributes which can be exploited. All pest growth functions are zero at zero pest. We also note that at worst for a pest control program, marginal pest growth rate at zero pest, that is the slope of the growth rate, is positive. We can consider, as a basic description for the pest growth for our model: f(0) = 0 and f0(0) 0.
Functional responses are also nil in the absence of pest (because there is no predation); they are always positive at positive pest numbers. So we can describe the prey-dependent functional response by these qualities: g(0) = 0 and 8x; g(x) 0. Intrapredator interfer-ence is perceived as a general decrease in the functional response as the predator population increase (see Buffoni et al. (2005), and Chapter 4 of this thesis). The numerical function is defined similarly, and so on.
Summary of model characteristics and aims
Throughout this thesis, we use qualitatively-defined impulsive models to describe augmenta-tive control programs. They are characterised by the four features below:
Bi-dimensionality. We assume that we are at the onset of a pest invasion and that crops or plants (the primary resource in our ecosystem) are in abundance. This means we make the assumption that the crop is implicit in the pest growth rate function. Unless stated otherwise, we consider that the predator species is a specialist, that is it feeds exclusively on the pest species. In the absence of the pest, the predators cannot sustain themselves and die (or leave the patch we model).
Implicit spatial effects. Second, we consider a homogeneous medium, the pest and preda-tor populations are uniformly distributed across the crop or in specific ‘local’ patches
– the size of which depend on the respective species mobility. Spatial effects such as migration to and from such patches are then implicitly taken into account in the net pest growth and predator mortality respectively else as an extension of the patch as its populations spread uniformly. This allows us to use a lumped-parameterisation deter-ministic formulation.
Table of contents :
.1 Understanding biological control
.1.1 Who kills who?
.1.2 Kill the pest!
.1.3 Use predators
.1.4 Augmentative control
.1.5 Predator types
.1.6 Integrated Pest Management (IPM)
.1.7 Some vital statistics
.2 Scope of this thesis
.3 Detailed plan
Making a choice: modelling approach
.1 Building the model
.1.1 Keeping count
.1.2 Some key assumptions
.1.3 Hybrid dynamics
.1.4 Canonical form
.2 Two mathematical properties of the impulsive model
.3 Why go impulsive?
.4 Why go qualitative? – I. Structural advantages
.4.1 Predator-prey processes
.4.2 Asymptotic behaviour
.5 Why go qualitative? – II. Field realities
.5.1 Finding the right compromise
.5.2 One model to rule them all
.6 Summary of model characteristics and aims
.6.1 Modelling features
Groundwork: a simple model and analytic methods
.1 A simple model
.2 Mathematical analysis of the simple model
.2.1 A local preview: with or without continuous control
.2.2 Positivity of the system and existence of the zero-pest solution
.2.3 Stability of the zero-pest solution
.2.4 A note on the computation of the global attractivity condition
.2.5 Interpreting the LAS and GAS conditions
.3 Pest control strategy
.3.1 The minimal predator release rate
.3.2 The period of release
Predators interfering with each other uring predation
.1 The premises
.2 The model
.3 The zero-pest solution
.3.1 Stability with affine parameterisation
.3.2 Stability with nonlinear parameterisation
.4 The minimal predator release rate
.4.1 Affine parameterisation
.4.2 Nonlinear parameterisation
.5 The pest evolution rate
.5.1 Affine parameterisation
.5.2 Nonlinear parameterisation
.6 The pest control strategy
.6.1 Choice of the predator species
.6.2 The predator releases
.6.3 Additional questions: beyond sector modelling
Cannibalism among predators
.1 Understanding cannibalism
.1.1 Cannibalism in the biomathematical jargon
.1.2 Impact of cannibalism on augmentative pest control
.1.3 General model of cannibalism
.2 Mechanism I: Territoriality
.2.1 The premises
.2.2 The model and specific hypotheses
.2.3 Existence of the zero-pest solution
.2.4 Stability of the zero-pest solution
.2.5 The minimal release rate
.3 Mechanism II: Hunger and fighting for survival
.3.1 The premises
.3.2 The model and specific hypotheses
.3.3 Existence of the zero-pest solution
.3.4 Stability of the zero-pest solution
.4 Mechanism III: Diet, alternate prey & intraguild predation
.4.1 The premises
.4.2 The model and specific hypotheses
.4.3 Existence and stability of the zero-pest solution
.4.4 The Kohlmeier- Ebenhöh model
.5 The pest control strategy
.5.1 Predator releases
.5.2 The selection of the predator species
Partial harvest effects
.1 The premises
.2 The model
.3 Existence of the zero-pest solution
.3.1 Releases more frequent than harvests
.3.2 Releases less frequent than harvests
.4 Stability of the pest-free solution
.4.1 The methodology
.4.2 Releases more frequent than harvests
.4.3 Releases less frequent than harvests
.5 The minimal release rate
.5.1 Dependence in biological processes
.5.2 Dependence in the period of release
.5.3 Dependence in partial harvesting parameters
.6 The pest control strategy
Practical guidelinesnd experimental results
.1.1 Matching data with the model
.1.2 Simple augmentative control program
.1.3 Interfering predators
.1.5 Integrated Pest Management (or partial harvests)
.2 Experiments with the predator Neoseiulus cucumeris to control the pest rankliniella occidentalis
.2.1 Experiment 1: intrapredator interference
.2.2 Experiment 2: release frequencies and pest control
Conclusion and prospects