The Consumer-Resource Relationship
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Claude Lobry is a former Professor at both the University of Bordeaux and the University of Nice in France. His present research interests are the mathematical theory of "nonstandard" differential systems and mathematical modeling of population dynamics.
Content
Preface ix
Chapter 1. History of the Predator-Prey Model 1
1.1. The logistic model 1
1.1.1. Notations, terminology 2
1.1.2. Growth with feedback and resource 4
1.1.3. Another interpretation of the logistic equation: the interference between individuals 9
1.1.4. (r, a)-model or (r,K)-model? 11
1.1.5. Historical notes and criticisms 14
1.2. The Lotka-Volterra predator-prey model 14
1.2.1. The model 14
1.2.2. Model analysis 15
1.2.3. Phase portrait and simulations 19
1.2.4. Historical notes and criticisms 20
1.3. The Gause model 24
1.3.1. The model 24
1.3.2. Model simulations 26
1.3.3. Historical notes and criticisms 29
1.4. The Rosenzweig-MacArthur model 31
1.4.1. The model 31
1.4.2. Analysis and simulations 32
1.4.3. Historical remarks and criticisms 35
1.5. The "ratio-dependent" model 38
1.5.1. Model analysis and simulations 38
1.5.2. Historical notes and criticisms 41
1.6. Conclusion 42
Chapter 2. The Consumer-Resource Model 43
2.1. The general model 43
2.1.1. General assumptions on the model 44
2.1.2. Properties 45
2.2. The "resource-dependent" model 47
2.2.1. Development of the Rosenzweig-MacArthur model 47
2.2.2. Analysis of the RMA model 52
2.2.3. Variants of the RMA model 59
2.3. The Arditi-Ginzburg "ratio-dependent" model 65
2.3.1. Development of the "RC-dependent" and "ratio-dependent" model 65
2.3.2. Analysis of RC and ratio-dependent models 68
2.3.3. Simulations of the ratio-dependent model 77
2.4. Historical and bibliographical remarks 83
Chapter 3. Competition 87
3.1. Introduction 87
3.2. The two-species competition Volterra model 89
3.2.1. Population 2 wins the competition 89
3.2.2. Population 1 wins the competition 90
3.2.3. Coexistence of both populations 91
3.2.4. Conditional exclusion 92
3.2.5. Interference 93
3.3. Competition and the Rosenzweig-MacArthur model 93
3.3.1. Equilibria of the competition RMA model 94
3.3.2. The exclusion theorem at equilibrium 96
3.3.3. The exclusion theorem and the Volterra model 99
3.4. Competition with RC and ratio-dependent models 100
3.4.1. Characteristics at equilibrium 100
3.4.2. Growth thresholds and equilibria of model [3.10] 102
3.4.3. Stability of coexistence equilibria 106
3.4.4. Criticism of RC and ratio-dependent competition models 109
3.4.5. Simulations 110
3.5. Coexistence through periodic solutions 119
3.5.1. Self-oscillating pair (x, y) 119
3.5.2. Adding a competitor 121
3.6. Historical and bibliographical remarks 123
Chapter 4. "Demographic Noise" and "Atto-fox" Problem 125
4.1. The "atto-fox" problem 125
4.2. The RMA model with small yield 126
4.2.1. Notations, terminology 128
4.2.2. The "constrained system" 130
4.2.3. Phase portrait of [4.3] when ¿d crosses the parabola "far away" from the peak 132
4.2.4. Phase portrait when ¿d crosses the parabola "close" to the peak 139
4.3. The RC-dependent model with small yield 148
4.4. The persistence problem in population dynamics 151
4.4.1. Demographic noise and the atto-fox problem 153
4.4.2. Sensibility of atto-fox phenomena 159
4.4.3. About the very unlikely nature of canard values 163
4.5. Historical and bibliographical remarks 165
Chapter 5. Mathematical Supplement: "Canards" of Planar Systems 169
5.1. Planar slow-fast vector fields 169
5.1.1. Concerning orders of magnitude 169
5.1.2. First approximation: the constrained system 172
5.1.3. Constrained trajectories 173
5.1.4. Constrained trajectories and "real trajectories" 175
5.2. Bifurcation of planar vector fields 183
5.2.1. System equivalence 184
5.2.2. Andronov-Hopf bifurcation 186
5.3. Bifurcation of a slow-fast vector field 190
5.3.1. A surprising Andronov-Hopf bifurcation 190
5.3.2. The particular case: p=0 193
5.3.3. Some terminology 201
5.3.4. Back to the initial model 202
5.3.5. The general case p ¿ 0 204
5.4. Bifurcation delay 212
5.4.1. Another surprising simulation 212
5.4.2. One more surprise 216
5.4.3. The Shiskova-Neishtadt theorem 219
5.5. Historical and bibliographical remarks 220
Appendices.225
Appendix 1. Differential Equations and Vector Fields 227
Appendix 2. Planar Vector Field 235
Appendix 3. Discontinuous Planar Vector Fields 241
Bibliography 253
Index 259
1
History of the Predator-Prey Model
Hasty readers who would like to only focus on the mathematical aspects can, if they wish, skip to the next chapter which is logically independent of this one. However, it seems to us that the history of a subject has learning virtues and it would be a shame not to enjoy them. Moreover, the term history used in the title is unsuitable. This is not a study of the emergence of concepts and models in the scientific context of their time as a real historical study would require, but more simply the presentation of mathematical models in chronological order of their appearance. These are the models:
- - the logistic model (1840);
- - the Lotka-Volterra model (1925);
- - the Gause model (1936);
- - the Rosenzweig-MacArthur model (1963);
- - the Arditi-Ginzburg model (1989).
We will merely make a brief remark on the emergence and reception of the model at the end of each section.
1.1. The logistic model
This section presents a few general ideas and some notations to be used throughout the book.
1.1.1. Notations, terminology
A population is a set of identical individuals.
- - Individuals can designate inert matter, such as, for example, carbon molecules dissolved in water, or living individuals such as bacteria or complex organisms such as fish or mammals.
- - By identical we mean that they are identical from a "certain point of view": from the point of view of chemistry, all carbon atoms are identical, but carbon 12 and carbon 14 differ in their atomic nucleus; bacteria belonging to the same species are considered identical as well as bacteria from two different species that have the same growth characteristics.
- - The size of a population at time t is a real number x(t), which suggests, of course, that the number of individuals is very large so that it can be reasonably represented by a continuous variable. If, for example, our population totals 11, 386, 749 individuals and we take one "million individuals" as unit size, then the size will be the real number (with six decimal places) 11.386749. We will come back to this topic.
The size of the population may be the total number of individuals or still the total mass (the mass of each individual multiplied by the number); in the latter case, this is referred to as biomass. The number or biomass divided by surface or volume is referred to as density. Resources encompass everything that is necessary for the growth of individuals in a given population: bacteria consume chemical substances; microalgae consume chemical substances and light; viruses "consume" bacteria. The growth rate of individuals, therefore the population growth rate, depends on the presence of various resources. Thereby, we can write:
where the function (s1, s2, · · ·, sp) ? µ(s1, s2, ., sp) is the growth rate. The variables (s1, s2, · · ·, sp) represent the quantities of available resources. We may assume that the function µ is increasing in each of its variables, but this is not necessarily the case: there are cases where the increase in resource has an inhibitory effect. The growth rate can be understood in two ways: taking mortality into account (or disappearance1) or not.
REMARK 1.1.- The sentence:
"the individual growth rate, thus the population growth rate (· · · )"
deserves all of our attention. What is the "growth rate of an individual"? In the case of a micro-organism it will be the speed with which its biomass increases, divided by its current biomass. Nonetheless, this growth rate, which is a property of individuals, is usually not directly measured. What is more often measured is the population growth, such as, for example, the growth in the diameter of a mold spot: this is the µ of the above expression. It is generally the tendency to identify individual growth rate with that of the population in accordance with the reasoning: if an organization grows by 1% in 1 minute, then the same happens for a population of 106 individuals, which will be true only if the 106 individuals of the population all have equal access to resources and is not necessarily always the case. For example, for mold, peripheral individuals have more efficient access to the substrate. As shown in this example, there is no reason to assume that the population growth rate is, in general, independent of the size x of the population. As a result, it follows that:
This is what will be done when we consider "density-dependent" models.
In many ecologically interesting situations, there is a limiting resource which means that all resources except one are in such excess that their possible variation does not affect the value of µ, which is tantamount to assuming that µ only depends on a single variable s (see Remark 3.1 about this topic); if, in addition, we assume a constant mortality rate, we have the model:
Assume that s is constant. In this model, either s is such that µ(s) > d and we have an exponential growth, or on the contrary, µ(s) < d and we have an exponential decay that leads to extinction. However, the population size acts on the resource as well as on mortality which means that there is no reason for either s or d to be constant. Let us consider some examples.
1.1.2. Growth with feedback and resource
1.1.2.1. Resource is space
Assume that the population is composed of "plant" individuals. The plants produce seeds that are randomly scattered over a given territory; only the seeds that land on a point of the territory that is not already colonized by a plant can germinate. Let:
- - the surface colonizable at time t be: s(t);
- - the size of the population be x(t), which can be assimilated to the occupied surface;
- - the total available surface be ST.
Let dt be a small increase in time. We can write:
to express that the number of seeds that germinate in the time interval dt is proportional to the number of seeds that travel in the atmosphere, thereby to the number of plants "x", proportional to the colonizable surface "s" and to the time period "dt"; ? is a constant that depends on chosen units. That is, if we evaluate the limit for dt 0, the differential equation is:
and, more simply, if we overlook including time:
Nevertheless, the colonizable surface is:
We thus have a feedback of the population on the resource that is of the form:
In the language of automatic control2, we are referring to static feedback versus dynamic feedback where s depends on x through a differential equation (see next section). If, in the growth equation of x, we replace s by its value according to x, we obtain the loop system (still in the language of automatic control):
which is the well-known "logistic" differential equation that can be rewritten as:
[1.1]after establishing r = ? ST and K = ST.
REMARK 1.2.- The above equation [1.1], as a mathematical object is defined for all x in R but, in the interpretation that is made thereof as a model of growth of plants on an island, it must be restricted to the interval [0, K] since the area occupied by plants is necessarily positive and smaller than the total area available K = S0.
This remark is not insignificant. Let us reconsider the same model for the same situation and let us introduce an "immigration". What do we mean?
- - Imagine that using a process, we are able to colonize a surface Im dt for a time period dt. Taking this assumption into account, the new model would be:
which as a differential equation is defined on all R+*. This equation has the globally asymptotically stable equilibrium: which is strictly greater than K; however, in our interpretation of the model, this value cannot be reached: x must remain less than or equal to K.
- - Imagine another situation: during a period dt, an amount Im dt of seeds reach the island; however, only the seeds which will land on the colonizable surface will germinate. In this case, we will write the model:
which still has K as a globally asymptotically stable equilibrium.
In the example we have just chosen, things are very simple and the risk of error is quite low. However, when the situation is a little more complicated, it is easy to make mistakes as discussed in the next section.
1.1.2.2. The resource is the substrate concentration
Let us picture micro-organisms (bacteria, yeast, plankton, etc.)...
