a primer of theoretical population ecology with rstevenmh/stevens_chap8_draft.pdf · m. henry h....

M. Henry H. Stevens

A Primer of TheoreticalPopulation Ecology with RSPIN Springer’s internal project number, if known

– Monograph –

January 4, 2008

SpringerBerlin Heidelberg NewYorkHongKong LondonMilan Paris Tokyo

Contents

1 Simple Density Independent Growth . . . . . . . . . . . . . . . . . . . . . . 11.1 An Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 A Very Specific Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Exploring Population Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 So what is the general rule? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Projecting population into the future . . . . . . . . . . . . . . . . 61.5 Graphing Discrete Geometric Growth . . . . . . . . . . . . . . . . . . . . . . 6

1.5.1 Direct calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5.2 E!ects of Initial Population Size . . . . . . . . . . . . . . . . . . . . 71.5.3 E!ects of Di!erent Per Capita Growth Rates . . . . . . . . . 91.5.4 Graphing Geomemtric Growth Using curve, and

stepfun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5.5 Doubling (and Tripling) Time . . . . . . . . . . . . . . . . . . . . . . . 121.5.6 Variable Growth Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 Average versus Simulated Population Growth . . . . . . . . . . . . . . . 161.6.1 The simulation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.7 Continuous Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.7.1 Relating ! and r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.8 Comments on Simple Density Independent Growth Models . . . 32Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Density-Independent Demography . . . . . . . . . . . . . . . . . . . . . . . . . 352.1 A hypothetical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1.1 Postbreeding vs. Prebreeding Censes . . . . . . . . . . . . . . . . . 472.2 Confronting Demographic Models with Data with

Bootstrapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2.1 An Example: Chamaedorea Palm Demography . . . . . . . . 48

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

VI Contents

3 Density Dependent Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.1 Discrete Density Dependent Growth . . . . . . . . . . . . . . . . . . . . . . . 58

3.1.1 E!ect of initial population size on growth dynamics . . . . 643.1.2 E!ects of " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.1.3 E!ects of rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2 Continuous Density Dependent Growth . . . . . . . . . . . . . . . . . . . . 743.2.1 A parsing and deparsing of the logistic model . . . . . . . . . 743.2.2 Stability of the logistic growth model . . . . . . . . . . . . . . . . 773.2.3 Dynamics of Continuous Logistic Growth . . . . . . . . . . . . . 80

3.3 Fitting Models to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.3.1 The role of resources in altering population

interactions within a simple food web . . . . . . . . . . . . . . . . 82Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4 Metapopulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.1 Building models of open populations . . . . . . . . . . . . . . . . . . . . . . . 101

4.1.1 Separating colonization and extinction . . . . . . . . . . . . . . . 1034.2 Parallels with logistic growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5 Lotka-Volterra Interspecific Competition . . . . . . . . . . . . . . . . . . 1095.1 Discrete time model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.1.1 Code for discrete logistic competition . . . . . . . . . . . . . . . . 1115.1.2 Exploring the discrete logistic growth . . . . . . . . . . . . . . . . 112

5.2 Continuous time model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.2.1 Isoclines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.2.2 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.3.1 Partial derivatives of growth rates . . . . . . . . . . . . . . . . . . . 1235.3.2 Solving the Jacobian at an equilibrium . . . . . . . . . . . . . . . 1245.3.3 Using Eigenanalysis to Assess Stability . . . . . . . . . . . . . . . 1275.3.4 Return time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.3.5 E!ects of r on stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6 Consumer-Resource Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.1 Lotka-Volterra predator-prey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.1.1 Long term dynamical behavior . . . . . . . . . . . . . . . . . . . . . . 1426.1.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466.1.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.2 Rosenzweig-MacArthur Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.2.1 Isoclines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.2.2 Rates of change with respect to each other: Partial

derivatives and the Jacobian matrix . . . . . . . . . . . . . . . . . 1546.2.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Contents VII

6.2.4 The paradox of enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 1576.3 Space, Hosts, and Parasitoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.3.1 Independent and random attacks . . . . . . . . . . . . . . . . . . . . 1606.3.2 Simulating Random Attacks . . . . . . . . . . . . . . . . . . . . . . . . 1626.3.3 Analytical Approximation of Random Attacks . . . . . . . . 1666.3.4 The Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1666.3.5 Aggregation leads to coexistence . . . . . . . . . . . . . . . . . . . . 167

6.4 Disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.4.1 Modeling Data from Bombay . . . . . . . . . . . . . . . . . . . . . . . 1796.4.2 Mass action transmission rate . . . . . . . . . . . . . . . . . . . . . . . 1826.4.3 Adding population dynamics . . . . . . . . . . . . . . . . . . . . . . . . 184

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

7 E!ects of Food Chain Length and Omnivory on FoodWeb stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897.2 Pimm and Lawton (1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1907.3 Implementing Pimm and Lawton 1977 . . . . . . . . . . . . . . . . . . . . . 1937.4 Shortening the Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1957.5 Adding Omnivory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

7.5.1 Comparing Chain A versus B . . . . . . . . . . . . . . . . . . . . . . . 1977.6 Re-evaluating Take-Home Messages . . . . . . . . . . . . . . . . . . . . . . . . 1987.7 Relations between Interaction Strength and Return Time? . . . . 200Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

8 Multiple Basins of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2038.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

8.1.1 Alternate stable states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2038.1.2 Multiple basins of attraction . . . . . . . . . . . . . . . . . . . . . . . . 204

8.2 Strong interference competition as a cause of multiple basinsof attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.2.1 Modeling competition with stochastic initial

population sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2068.2.2 Resource competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.3 Intraguild Predation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2178.3.1 The simplest Lotka-Volterra model of IGP . . . . . . . . . . . . 2178.3.2 A Lotka-Volterra model of IGP with resource

competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2188.3.3 E!ects of relative abundance . . . . . . . . . . . . . . . . . . . . . . . . 2218.3.4 E!ects of absolute abundance . . . . . . . . . . . . . . . . . . . . . . . 2238.3.5 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

VIII Contents

9 Competition, Colonization, and Finite Rates of Succession . 2319.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2319.2 Competition-Colonization Tradeo! . . . . . . . . . . . . . . . . . . . . . . . . 231

9.2.1 Multispecies competition-colonization tradeo! . . . . . . . . 2369.2.2 Habitat Destruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

9.3 Adding Reality: Finite Rates of Competitive Exclusion . . . . . . . 2429.4 The Successional Niche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

9.4.1 Mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2499.4.2 Competitive exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

A A Brief Introduction to R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253A.1 Strengths of R/S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253A.2 The R Graphical User Interface (GUI) . . . . . . . . . . . . . . . . . . . . . 254A.3 Where is R? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256A.4 Starting at the Very Beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

B Programming in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259B.1 Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259B.2 Simple Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

B.2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260B.2.2 Getting information About Vectors . . . . . . . . . . . . . . . . . . 261B.2.3 Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

B.3 Missing Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263B.4 Matrices and Data Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

B.4.1 Extraction in Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265B.4.2 Data Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

B.5 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268B.5.1 Data frames are also lists . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

B.6 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269B.6.1 Writing your own functions . . . . . . . . . . . . . . . . . . . . . . . . . 271

B.7 Sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272B.8 Iterated Actions: “apply” and loops . . . . . . . . . . . . . . . . . . . . . . . . 273

B.8.1 Iterations of independent actions . . . . . . . . . . . . . . . . . . . . 273B.8.2 Dependent iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

B.9 Rearranging and aggregating information in data frames . . . . . 275B.9.1 Rearranging or reshaping data . . . . . . . . . . . . . . . . . . . . . . 275B.9.2 Summarizing by groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

B.10 Getting Data Out Of and Into the Workspace . . . . . . . . . . . . . . . 282B.11 Probability Distributions and Randomization . . . . . . . . . . . . . . . 282B.12 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284B.13 Numerical Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286B.14 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288B.15 Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

B.15.1 plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288B.15.2 Adding points, lines and text to a plot . . . . . . . . . . . . . . . 289

Contents IX

B.15.3 More than one response variable . . . . . . . . . . . . . . . . . . . . 290B.15.4 Controlling Graphics Devices . . . . . . . . . . . . . . . . . . . . . . . 291B.15.5 Creating a Graphics File . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

B.16 Graphical displays that show distributions . . . . . . . . . . . . . . . . . . 292

C Stability and Eigenanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297C.1 Stability of Systems of Di!erential Equations . . . . . . . . . . . . . . . 297

C.1.1 The shortest version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297C.2 The short version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

C.2.1 Further explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298C.3 Eigenanalysis of demographic versus Jacobian matrices . . . . . . . 300

8

Multiple Basins of Attraction

Summary

In this chapter we explore how the dynamics of simple, and perhaps com-mon, food webs are sensitive to history or initial conditions. In the process, Ihope you gain a greater appreciation of distinctions, or lack there of, betweentransient and equilibrial behavior of a system.

8.1 Introduction

Historically, simple models may have helped to lull ecologists into thinkingthinking either that models are useless because they do not reflect the naturalworld, or that the natural world is highly predictable.1 Here we investigatehow unpredictable outcomes derive in two very simple situations a competitivecommunity containing only three species, and also a three species food webof intraguild predation. In both cases, we get totally di!erent outcomes, oralternate stable states, depending on di!erent, stochastic, initial conditions.

8.1.1 Alternate stable states

Alternate stable states are a set of possible stable outcomes that can occurgiven a single set of conditions. For a single list of species, there may existmore than one combination of abundances that provide stable equilibria. Thiscombination of abundances may result from demographic stochasticity, dif-ferent assembly sequences, disturbance, or the gradual change and return ofan environmental factor. The last is referred to as hysteresis. The key is thatthe external environmental conditions must be the same; otherwise it is just

1 One notable exception was May’s work revealing that chaos erupts from a verysimple model of discrete logistic growth.

204 8 Multiple Basins of Attraction

a di!erent system. Note, however, that if an abiotic factor is coupled dynam-ically to the biotic community, then it is an internal part of the system andnot external to it.

Part of our thinking about alternate stable states relies on two assumptionsthat I and some others [90] find untenable, or at least not very useful. First, theconcept assumes that fixed stable states exist. Second, the concept assumesthat communities move to these states with all due haste, and achieve thesestates over observable time periods. If these conditions are met, then ASS’scan arise and be observed. I would argue that this is a very restricted, specialcase of a more general framework that has been referred to as multiple basinsof attraction.

8.1.2 Multiple basins of attraction

Multiple basins of attraction (MBA) is a phrase describing an entire landscapeof “tendencies” in community dynamics. Imagine for a moment a two or threedimensional coordinate system in which each axis is the abundance of onespecies. An attractor is merely a place in the coordinate system, that is, aparticular set of species abundances, that exerts a pull on, or attracts, thedynamics of the populations. A stable equilibrium is an example of a globalattractor; each local minimum or maximum in a stable limit cycle is also anattractor. The unstable equilibrium we observed in a Lotka-Volterra compe-tition model is another example — it is a saddle, or an attractor-repellor,because it attracts from one direction, but repels in another.

MBA’s can be visualized as a topographic landscape, or mountain range.We see lots of little valleys (attractors) and lots of peaks (repellors) (Fig. 8.1).If we think more broadly about multiple basins of attraction, then we beginto see that a strict definition of ASS lies at one end of a continuum, and itis matched at the other end by a system with one global stable equilibrium.The entire continuum, and much more besides, can be conceived of in termsof a landscape with some number of both basin of attraction (attractors) andpeaks, which are repellors in the landscape; each of these may act with someunique force or strength.

The potential for MBA’s to occur has been noted for a few decades [91, 92,93, 94, 95], but the hunt for their signature in nature has begun more recently[96, 97, 98, 99, 100, 101, 102, 103]. There has been discussion about whatcauses more than one basin of attraction, including space, stochasticity, andpredation. Here we examine two mechanisms that may reveal priority e!ects:strong interference competition [104], and intraguild predation, an indirectinteraction which has elements of both competition and predation [105].

8.2 Strong interference competition as a cause of multiple basins of attraction 205

Fig. 8.1: Perspective and contour plots of a single complex dynamical landscape,containing multiple basins of attraction. Imagine putting a ball into this landscape,and jiggling the landscape around. The landscape represents the possible states of thecommunity, the ball represents the actual community structure at any one point intime, and the jiggling represents stochasticity, either demographic, or environmental.

8.2 Strong interference competition as a cause ofmultiple basins of attraction

You have already seen in lecture the case of two species Lotka-Volterra com-petition where an attractor-repellor, or saddle, arises from the conditions foran unstable equilibrium. This arises when interspecific competition is greaterthan intraspecific competition. When this is the case, each species can sup-press the other, if it gets a head start.

Having a larger negative e!ect on your competitor than on yourself maynot be too unusual. Examples that come immediately to mind are cases wheremembers of one population cooperate, but compete against members of adi!erent population, such as two clonal plant species, social animals. Likewise,if species compete preemptively for space (territories, or substrate surface) orfor a resource with a unidirectional flow (drifting prey in a stream, or lightcoming down on a forest canopy), then one species may gain the upper handby preempting resources.

In human economic systems, businesses can have direct negative e!ects oneach other through unfair business practices. For instance, a larger companycan temporarily flood a local market with below-cost merchandise, and driveout smaller competitors, prior to raising prices again. In addition, economiesof scale may give larger businesses an upper hand.

Here we explore how MBA can arise in a competitive system. We use athree species Lotka-Volterra model to illustrate how strong interference com-petition may reveal priority e!ects. We use a slightly di!erent representation


of the Lotka-Volterra competition model. All "’s are per capita e!ects of onepopulation on it’s own or others’ growth rates.

dN1

dt= r1N1 (1! "11N1 ! "12N2 ! "13N3) (8.1)

dN1

dt= r2N2 (1! "21N1 ! "22N2 ! "23N3) (8.2)

dN1

dt= r3N3 (1! "31N1 ! "32N2 ! "33N3) (8.3)

Note three aspects of the above equations. First note that within the paren-theses, the N ’s are all arranged in the same order. This reflects their relativepositions in a food web matrix, and reveals the row-column relevance of thesubscripts of the "’s. Second, note that "ii = 1/Ki, and in some sense Ki

results from a particular "ii. Third, note that the "ij represent the simple percapita e!ect of species j on species i, rather than the ratio of two per capitae!ects typically described in text books that use the ", #, and K notation.

Here we create a function for these ODEs, taking advantage of matrixoperations. Note that we can represent the three species as we would one,dN/dt = rN (1! "N), because the "N actually becomes a matrix operation,a %*% N.2

> LVComp3 <- function(t, n, parms) {+ r <- parms[[1]]+ a <- parms[[2]]+ dns.dt <- r * n * (1 - (a %*%+ n))+ return(list(c(dns.dt)))+ }

We are going to use one set of parameters, but let initial abundances varystochastically around the unstable equilibrium point, and examine the results.

8.2.1 Modeling competition with stochastic initial population sizes

Next we decide on the values of the parameters. We will create intrinsic ratesof increase, r’s, and intraspecific competition coe"cients, "ii that correspond,roughly to an r!K tradeo!, that is, between maximum relative growth rate(r) and carrying capacity (1/"ii). Species 1 has the lowest maximum relativegrowth rate and the weakest intraspecific density dependence.

Following these ecological guidelines, we create a vector of r’s, and a matrixof "’s. We then put them together in a list,3 and show ourselves the result.

2 Recall that %*% is matrix multiplication in R because by default, R multipliesmatrices and vectors elementwise.

3 A list is a specific type of R object.


> r <- c(r1 = 0.6, r2 = 1, r3 = 2)> a <- matrix(c(a11 = 0.001, a12 = 0.002,+ a13 = 0.002, a21 = 0.002, a22 = 0.00101,+ a23 = 0.002, a31 = 0.002, a32 = 0.002,+ a33 = 0.00102), nrow = 3, ncol = 3)> parms <- list(r, a)> parms

[[1]]r1 r2 r30.6 1.0 2.0

[[2]][,1] [,2] [,3]

[1,] 0.001 0.00200 0.00200[2,] 0.002 0.00101 0.00200[3,] 0.002 0.00200 0.00102

Next we get ready to simulate the populations 24 times. We set the time, t,and the mean and standard deviation of the initial population sizes. We thencreate a matrix of initial population sizes, with one set of three species’ n0 foreach simulation. This will create a 3" 24 matrix, where we have one row foreach species, and each column is one of the initial sets of population sizes.

> t = seq(0, 40, by = 0.1)> ni <- 200> std = 10> N0 <- sapply(1:30, function(i) rnorm(3,+ mean = ni, sd = std))

Now let’s replace the first set of initial abundances to see what would happenif they start out at precisely the same initial abundances. We can use that asa benchmark.4

> N0[, 1] <- ni

When we actually do the simulation, we get ready by first creating a graphicsdevice (and adjust the margins of each graph). Next we tell R to create agraph layout to look like a 6 " 4 matrix of little graphs. Finally, we runthe simulation, calling one column of our initial population sizes at a time,integrate with the ODE solver, and plot the result, 24 times. As we plot theresult, we also record which species has the greatest initial abundance.

Now consider the output (Fig. 8.2.1). Recall that these species di!er intheir intrinsic rates of increase (r); those with higher r have slightly greaternegative intraspecific density dependence. They all, however, have twice a

4 R’s recycling rule tells it to use the single value of ni for all three values in thefirst column of N0.


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Fig. 8.2: Interaction between strong interference competition and initial abundanceswith three competing species. Interspecific competition is approximately twice thatof intraspecific competition (r1 = 0.6, r2 = 1, r3 = 2, a11 = 0.001, a22 = 0.00101,a33 = 0.00102, aij,i!=j = 0.002). Solid line, species 1; dashed line, species 2; dot-ted line, species 3; on each plot we indicate the species with the greatest initialabundance.

big a negative e!ect on each other as they do on themselves. Species 1 hasthe highest carrying capacity, and would therefore often considered the bestcompetitor. Note, however, that when they start at equal abundances (Fig.8.2.1, upper left corner), the species with the intermediate carrying capacityand intermediate r displaces the other two species. We don’t go into it here,but when three or more species are included in a Lotka-Volterra competitive


community, coexistence depends on both r and on the "’s. This contrasts tothe two species case where coexistence is typically thought to depend only onthe "’s (but see [106]).

What can you detect in Fig. 8.2.1? Two species never coexist, but do yousee each species win at least once? For which species do you see it start at thehighest initial abundance, and yet not win? The points here is that (i) witha little stochasticity in initial conditions, this simple model generates unpre-dictable outcomes, and (ii) initial abundance does not determine everything.

It is critical to realize that this is occurring because species have largernegative competitive e!ects on others than they have on themselves. In thiscase the e!ects are direct, because the model is Lotka-Volterra competition.The e!ects may also be indirect, when species compete for more than onelimiting resource. MacArthur [57] showed, for instance, that when generalistsand specialists compete so that not all resources are available to all species,alternate stable states occur; Long and Karel [97] showed just such a result.Next, we take on just such an example.

8.2.2 Resource competition

Above, we explored how simple Lotka-Volterra competition could result inunstable equilibria, causing saddles, and multiple basins of attraction. Herewe take a look an example of how resource competition can do the samething. Recall that resource competition is really an indirect interaction, wherespecies interact through shared resources.

Sche!er and colleagues [101] provide evidence that anthropogenic enriched(eutrophic) lakes can shift away from dominance by submerged plants, rootedin substrate, into systems completely dominated by floating plants such asduckweed (Lemna spp.) and water fern (Azolla spp.). Submerged macro-phytes5 can extract nutrients out of both sediments and the water column. Atlow, typically unpolluted, nutrient levels, submerged plants can draw downwater nitrogen levels to a very low level, below levels tolerated by duckweedand water fern. At high nutrient levels, floating plants are no longer limited bywater column nitrogen levels, and can create deep shade that kills submergedvegetation. Aside from killing these wonderful submerged macrophytes, thisshift typically alters the rest of the lake food web.

Sche!er and colleagues represented these interactions in the following man-ner, where F and S are floating and submerged plants, respectively.

dF

dt= rfF

n

n + hf

11 + afF

! lfF (8.4)

dS

dt= rsS

n

n + hs

11 + asS + bF + W

! lsS (8.5)

(8.6)

5 “Macrophyte” is a term often used for vascular plants


As usual, r represents the maximum per capita rate of increase for F and Srespectively. This maximum is modified by three di!erent expressions,

Nitrogen limitation The first factor to modify maximum growth rate in eq.8.4 is n/ (n + hx), water column nitrogen limitation, and varies from 0–1;when h = 0 there is no limitation and the fraction equals 1.0 at all nutrientlevels. If h > 0, then the growth rate is a Michaelis-Menten type saturatingfunction where the fraction approaches zero as n# 0, but increases toward1.0 (no limitation) at high nutrient levels. For submerged plants, hs = 0because they are never really limited by water column nitrogen levelsbecause they derive most of the nitrogen from the substrate.

Light limitation The second factor to modify maximum growth rate is lightlimitation; 1/af and 1/as are half-saturation constants — they determinethe plant densities at which the growth rates are half of the maxima; brepresents the shade cast by floating plants, and W represents the lightintercepted by the water column.

Loss The second terms in the above expressions lfF and lsS are simplydensity independent loss rates due to respiration or mortality.

Nitrogen limitation is a simple saturating function that declines with increas-ing plant biomass which achieves a maximum N in the absence of any plants.

n =N

1 + qsS + qfF(8.7)

The nutrient concentration, n, depends not only on the maximum N nutrientconcentration, but also on the e!ect of submerged and floating plants whichtake up nitrogen out of the water at rates 1/ (1 + qSS) and 1/ (1 + qfF ) re-spectively; 1/q is is the half-saturation constant.

Sche!er and colleagues very nicely provide units for their parameters andstate variables (Table 8.2.2).

Table 8.1: Parameter and variable units and base values. Plant mass (g) is dryweight.

Parameter/Variable Value UnitsF , S (varies) gm"2N , n (varies) mgL"1as, af 0.01 (gm"2)"1b 0.02 (gm"2)"1qs, qf 0.075, 0.005 (gm"2)"1hs, hf 0.0, 0.2 mgL"1ls, lf 0.05 g g"1 day"1rs, rf 0.5 g g"1 day"1

Consider the meanings of the parameters (Table 8.2.2). What is a, andwhy is 1/as = 1/af? This indicates that both plants become self-shading at


the same biomasses. Why is qs > qf? This indicates that a gram of submergedplants can pull more nitrogen out of the water than a gram of floating plant.Last, why is hs = 0? Because submerged plants grow independently of thenitrogen content in the water column.

Now we are set to model this in R.

> Scheffer.ode <- function(t, y,+ p) {+ F <- y[1]+ S <- y[2]+ with(as.list(p), {+ n <- N/(1 + qs * S + qf *+ F)+ dF <- rf * F * (n/(n ++ hf)) * (1/(1 + af *+ F)) - lf * F+ dS <- rs * S * (n/(n ++ hs)) * (1/(1 + as *+ S + b * F + W)) - ls *+ S+ return(list(c(dF, dS)))+ })+ }

Now let’s set the parameters (Table 8.2.2), time, initial abundances, and seewhat we have.

> p <- c(N = 1, as = 0.01, af = 0.01,+ b = 0.02, qs = 0.075, qf = 0.005,+ hs = 0, hf = 0.2, ls = 0.05,+ lf = 0.05, rs = 0.5, rf = 0.5,+ W = 0)> t <- 1:200> Initial <- c(F = 10, S = 10)> S.out1 <- lsoda(Initial, t, Scheffer.ode,+ p)

> matplot(t, S.out1[, -1], type = "l")> legend("right", c("F", "S"), lty = 1:2,+ bty = "n")

From this run, at these nutrient levels, we observe the competitive dominanceof the submerged vegetation (Fig. 8.4a). Let’s increase nitrogen and see whathappens.

> p["N"] <- 4> S.out2 <- lsoda(Initial, t, Scheffer.ode,+ p)


> matplot(t, S.out2[, -1], type = "l")

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Fig. 8.3: The outcome of competition depends on the nutrient loading.

Ah-ha! At high nutrient levels, submerged vegetation triumphs (Fig. 8.4b).So where are the cool multiple basins of attraction? Let’s mimic nature by let-ting the e!ect of Homo sapiens increasing gradually, with gradually increasingover-exploitation of the environment. We will vary N , increasing it slowly, andhang on to only the final, asymptotic abundances, at the final time point.

> N.s <- seq(0.5, 4, by = 0.1)> t <- 1:1000> S.s <- t(sapply(N.s, function(x) {+ p["N"] <- x+ lsoda(Initial, t, Scheffer.ode,+ p)[length(t), 2:3]+ }))

Now we plot, not the time series, but rather the asymptotic abundances vs.the nitrogen levels (Fig. 8.4a).

> matplot(N.s, S.s, type = "l")> legend("right", c("F", "S"), lty = 1:2,+ bty = "n")

Now we can see this catastrophic shift at around 2.7 mg N L!1 (Fig. 8.4a). Asnitrogen increases, we see a gradual shift in community composition, but then,


wham!, all of a sudden a small additional increase at 2̃.7, we get dominanceby floating plants, and the loss of our submerged plants.

Now let’s try to fix the situation by reducing nitrogen levels. We mightimplement this by asking upstream farmers to use no-till practices, for instance[107]. This is equivalent too starting at high floating plant abundances, lowsubmerged plant abundances, and then see what happens at di!erent nitrogenlevels.

> Initial.Eutrophic <- c(F = 600,+ S = 10)> S.s.E <- t(sapply(N.s, function(x) {+ p["N"] <- x+ lsoda(Initial.Eutrophic, c(1,+ 1000), Scheffer.ode, p)[2,+ 2:3]+ }))

Now we plot, not the time series, but rather the asymptotic abundances vs.the nitrogen levels (Fig. 8.4b).

> matplot(N.s, S.s.E, type = "l")

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S

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(b) High Initial N

Fig. 8.4: The outcome of competition depends on the history of nutrient loading.

What a second! If we start at high floating plant biomass (Fig. 8.4b), thecatastrophic shift takes place at a much lower nitrogen level. This is telling usthat from around N =0.9–2.7, the system has two stable basins of attraction,


or alternate stable states. It might be dominated either by floating plants orby submerged plants. This is often described as hysteresis, where there is morethan one value of the response for each value of the predictor variable.

Hysteresis is usually represented as in Fig. 8.5, and it bears some expla-nation. Hysteresis is usually defined as a context-dependent response to anenvironmental driver. Imagine that the state variable in Fig. 8.5 is annualprecipitation, and the driver is average annual temperature. Imagine thatover many years, the regional average temperature increases, perhaps due toincreased CO2 in the atmosphere. At first, increased temperature has littlee!ect on precipitation. Once precipitation reaches a particular threshold, pre-cipitation drops dramatically, and additional increased temperature has littlee!ect. When the temperature starts to come back down, however, we find thatthe high levels of precipitation do not return at the same threshold where welost it. Rather, it does not return until we bring temperature way back downto original levels. At intermediate temperatures (grey region, Fig. 8.5), precip-itation depends on what the temperature used to be. This history-dependence,or context-dependence is the hallmark of hysteresis.6 It is important to under-stand that, in principle, this is not the result of a time lag. It is not the casethat this pattern is due to a lagged or slow response by the state variable.Rather, these alternate basins (Fig. 8.5, solid lines in the grey area) representpermanent stable states from which the response variable cannot ever emerge,without some external force, such as very large changes in the environmentaldriver. Time lags may be important in other, di!erent, circumstances, but arenot, in principle, related to hysteresis.

Now let’s represent these changes for the two state variables in the aquaticplant model. First we represent the floating plants. Here we plot the lowabundance state for the floating plants, adjusting the figure margins to ac-commodate all abundances, and then add in the high abundance data (Fig.8.6a).

> plot(N.s[1:23], S.s[1:23, 1], type = "l",+ lwd = 2, xlim = c(0, 4), ylim = c(0,+ 900), main = "Floating Plants",+ ylab = expression("Biomass (g m"^-2 *+ ")"), xlab = "Nitrogen Supply Rate")> lines(N.s[-(1:5)], S.s.E[-(1:5),+ 1], lwd = 2)

Here we reinforce the concepts of multiple basins and hysteresis, by showingwhere the attractors are. I will use arrows to indicate these basins. At eitherhigh nitrogen or very low nitrogen, there is a single, globally stable attractor.At low nutrients, only submerged plants exist regardless of starting conditions.

6 Sometimes the response depends on what the driver used to be, and sometimesit depends on what the state variable itself used to be; that distinct depends onthe links between the driver and the state variable.


Environmental Variable(e.g. Temperature)

Stat

e Va

riabl

e(e

.g. P

recip

itatio

n)

Fig. 8.5: Hysteresis. When the system changes from left to right, the threshold valueof the predictor of catastrophic change is greater than when the system moves fromright to left. At intermediate levels of the driver, the value of the response variabledepends on the history of the system. Each value, either high or low, represents analternate, stable, basin of attraction. The arrows in this figure represent the directionof change in the environmental driver.

At high nutrients, only floating plants persist. Let’s put in those arrows (Fig.8.6a).

> arrows(3, 10, 3, 620, length = 0.1)> arrows(3, 820, 3, 720, length = 0.1)> arrows(0.5, 620, 0.5, 50, length = 0.1)

Next we want arrows to indicate the alternate basins of attraction at interme-diate nitrogen supply rates. Floating plants might be at kept at low abundanceat intermediate nitrogen supply rates if submerged plants are abundant (Fig.8.4b). Let’s indicate that with a pair of arrows.

> arrows(2.5, -10, 2.5, 60, length = 0.1)> arrows(2.5, 200, 2.5, 100, length = 0.1)> text(2.5, 100, "Coexisting\nwith S",+ adj = c(1.1, 0))

Alternatively, if submerged plants were at low abundance, floating plantswould get the upper hand by lowering light levels, which would exclude sub-merged plants altogether (Fig. 8.6a). Let’s put those arrows in (Fig. 8.6a).


> arrows(2, 480, 2, 580, length = 0.1)> arrows(2, 750, 2, 650, length = 0.1)> text(2, 700, "Monoculture", adj = c(1.1,+ 0))

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ass

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ass

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Fig. 8.6: The outcome of competition depends on the nutrient loading.

Now let’s repeat the exercise with the submerged plants. First we plot thehigh abundance state, and then add the low abundance state (Fig. 8.6b).

> plot(N.s[1:23], S.s[1:23, 2], type = "l",+ lwd = 2, xlim = c(0, 4), ylim = c(0,+ 900), main = "Submerged Plants",+ ylab = expression("Biomass (g m"^-2 *+ ")"), xlab = "Nitrogen Supply Rate")> lines(N.s[-(1:5)], S.s.E[-(1:5),+ 2], lwd = 2)

Now we highlight the global attractors that occur at very low or very highnitrogen supply rates (Fig. 8.6b).

> arrows(0.7, 30, 0.7, 830, length = 0.1)> arrows(3.8, 830, 3.8, 30, length = 0.1)

Next we highlight the local, alternate stable equilibria that occur at interme-diate nitrogen supply rates; either the submerged plants are dominating dueto nitrogen competition, and achieving high abundance,

8.3 Intraguild Predation 217

> arrows(2.3, 650, 2.3, 750, length = 0.1)> arrows(2.3, 900, 2.3, 800, length = 0.1)> text(2.4, 900, "Coexisting\nwith F",+ adj = c(0, 1))

or they are excluded entirely, due to light competition (Fig. 8.6b).

> arrows(2, 130, 2, 30, length = 0.1)> text(2, 140, "Excluded\nDue to Light Comp.",+ adj = c(0.5, -0.3))

Once again, we see what underlies these alternate states, or basins. Onepopulation gains a numerical advantage that results in an inordinately largenegative e!ect on the loser, and this competitive e!ect comes at little cost tothe dominant species. At intermediate nitrogen supply rates, the submergedvegetation can reduce ambient nitrogen levels in the water column to unde-tectable levels because it gets most of its nitrogen from sediments. On theother hand, if floating plants can ever achieve high densities (perhaps due toa temporary nutrient pulse), then the shade they cast at intermediate supplyrates prevents lush growth of the submerged plants. As a consequence, thesubmerged plants can never grow enough to drawn nitrogen levels down toreduce the abundance of the floating plants.

8.3 Intraguild Predation

Intraguild predation di!ers from omnivory only in degree. In omnivory, apredator shares a resource with one or more of its prey (Fig. 8.3). Thus thetop predator feeds relatively high on the food chain, getting most of its energyor resources by eating its competitor (a > 0.5 in Fig. 8.3). An extension ofthis is the case of intraguild predation, in which a species preys upon one ormore of its competitors (Fig. 8.3). Intraguild predation is thus refers to thecase in which the top predator gets most of its energy or resources from themore basal resource, eating lower on the food chain (a < 0.5 in Fig. 8.3). Thedistinction is not qualitative, but rather quantitative. If both consumer speciesprey upon each other, then we could make the argument that the name weascribe to it depends entirely upon one’s perspective. In such a case, however,we generally refer to the relations as intraguild predation.

8.3.1 The simplest Lotka-Volterra model of IGP

We can extend our good ol’ Lotka-Volterra competition model to describeintraguild predation. Let’s do that, just for kicks. All we do is add a termonto each competitor, where one eats, and one gets eaten.


N

P

a

B

1-a A

Fig. 8.7: We typically use “omnivory” when a > 0.5, and “intraguild predation”a < 0.5). If we remove A from this model, then the species represent those of Holtand Polis (1997).

dN1

dt= r1N1 (1! "11N1 ! "12N2) + baN1N2 (8.8)

dN2

dt= r2N2 (1! "21N1 ! "22N2)! aN1N2 (8.9)

Here a is attack rate of the IGP predator, N1, on the prey, N2; b is theconversion e"ciency of the prey into predator growth. You will notice that inthis model, the predator has a type I functional response, simple mass action.

Let’s work through a little logic.

• Each species will achieve its usual carrying capacity, 1/"ii, in the absenceof the other.

• If we have stable coexistence, then adding predation will increase the riskof extinction for the prey, and increase the abundance (if only temporarily)for the predator.

• If the poorer competitor is able to feed on the better competitor, this hasthe potential to even the scales.

• If the poor competitor is also the prey, then — forget about it — thechances of persistence by the competitor are poor indeed.

Now let’s move on to a model of intraguild predation with resource com-petition.

8.3.2 A Lotka-Volterra model of IGP with resource competition

Here we introduce a simple IGP model from Holt et al. [108], where the com-petition between consumers is explicit resource competition rather than directcompetition as above. The resource for which they compete is a logistic pop-ulation.


dP

dt= #PB"BP PB + #PN"NP PN !mP P (8.10)

dN

dt= #NB"BNBN !mNN ! "NP PN (8.11)

dB

dt= rB (1! "BBB)! "BNBN ! "BP PB (8.12)

Recall that the units for attack rate, ", are number of prey (killed) per indi-vidual of prey per individual of predator; the units for conversion e"ciency, #,are number of predators (born) per number of prey (killed, and presumablyeaten and assimilated). The consumers in this model have a type I functionalresponse (mass action), and negative density dependent population growth inthe basal species.

Holt et al. show analytically that five equilibria are present [108] .1. all species have zero density,2. only the resource, B, is present, at B = K,3. only the resource, B, and IG-prey, N , are present.4. only the resource, B, and IG-predator, P , are present.5. all species present.

We will explore how initial conditions influence the outcomes of this simpleIGP model. We will focus on the last three equilibria, with two or three speciespresent.

We start by creating an R function for the above Lotka-Volterra intraguildpredation model.

> IGP.ODE <- function(t, y, params) {+ B <- y[1]+ N <- y[2]+ P <- y[3]+ with(as.list(params), {+ dPdt <- bpb * abp * B *+ P + bpn * anp * N *+ P - mp * P+ dNdt <- bnb * abn * B *+ N - mn * N - anp *+ N * P+ dBdt <- r * B * (1 - abb *+ B) - abn * B * N -+ abp * B * P+ return(list(c(dBdt, dNdt,+ dPdt)))+ })+ }

In this code, I used three-letter abbreviations ("NP = anp). The first letter,a or b, stands for " and #. The next two lower case letters correspond to oneof the populations, B, N , and P .


Next, we create a vector to hold all those parameters.

> params1 <- c(bpb = 0.032, abp = 10^-8,+ bpn = 10^-5, anp = 10^-4, mp = 1,+ bnb = 0.04, abn = 10^-8, mn = 1,+ r = 1, abb = 10^-9.5)

Here we get ready to actually do the simulations or numerical integration withlsoda. We set the time, and then we set four di!erent sets (rows) of initialpopulation sizes, label them, and look at them.

> t = seq(0, 60, by = 0.1)> N.init <- cbind(B = rep(10^9, 4),+ N = 10^c(2, 5, 3, 4), P = 10^c(5,+ 2, 3, 4))

Now we integrate the population dynamics and look at the results. Here wefirst set up a graphics device with a layout of four figures and fiddle with themargins. We then use a for-loop to integrate and plot four times. Then weadd a legend.

> quartz(, 4, 4)> layout(matrix(1:4, nr = 2))> par(mar = c(4, 4, 1, 1))> for (i in 1:4) {+ igp.out <- lsoda(N.init[i,+ 1:3], t, IGP.ODE, params1)+ matplot(t, log10(igp.out[,+ 2:4] + 1), type = "l",+ lwd = 2, ylab = "log(Abundance)")+ }> legend(40, 9, c("B", "N", "P"),+ lty = 1:3, col = 1:3, lwd = 2,+ bty = "n")

Clearly, initial abundances a!ect which species can coexist (Fig. 8.8). Ifeither consumer begins with a big advantage, it excludes the other consumer.In addition, if they both start at low abundances, the IGP prey, N , excludesthe predator; if they start at moderate abundances, the IGP predator, P ,wins.

Now we need to get more thorough and systematic. The above code andits results show us the dynamics (through time) of particular scenarios. This isgood, because we need to see how the populations change through time, just tosee if funky things happen, because sometimes unexpected dynamics happen.A complementary way to analyze this model is to vary initial conditions moresystematically and more thoroughly, and then simply examine the end points,rather than each entire trajectory over time. It is a tradeo! — if we want tolook at a lot of di!erent initial conditions, we can’t also look at the dynamics.


0 20 40 60

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8

tlog(Abundance)

0 20 40 60

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8

t

log(Abundance)

0 20 40 60

24

68

t

log(Abundance)

0 20 40 60

02

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8

tlog(Abundance) B

NP

Fig. 8.8: Dynamics of Lotka Volterra intraguild predation, with di!ering initial abun-dances. See code for parameter values.

In the next sections, we examine the e!ects of relative abundance of thetwo consumers, and then of their absolute abundances.

8.3.3 E!ects of relative abundance

First we will vary the relative abundances of the IGP prey and predator, Nand P . We create a slightly more complete set of initial abundances, with Bconstant, and N increases as P decreases.

> logNP <- seq(2, 5, by = 0.1)> N.inits <- cbind(B = rep(10^9,+ length(logNP)), N = 10^logNP,+ P = 10^rev(logNP))

We see (scatterplot matrix not shown) that we do have negative covariationin the starting abundances in the two consumer species, the IG-prey and IG-predator.


Next, we need to perform all7 the simulations, and hold on to all the end-points.8 We do it over a long time span to see the (hopefully) truly asymptoticoutcomes. We use a little manipulation to hang on to the initial abundances,at t = 50 and the final abundances at t = 500, putting them each in their owncolumn and hanging on to it.

> t1 <- 1:500> MBAs <- t(sapply(1:nrow(N.inits),+ function(i) {+ tmp <- lsoda(N.inits[i,+ ], t1, IGP.ODE, params1,+ hmax = 0.1)+ cbind(tmp[1, 3:4], tmp[50,+ 3:4], tmp[500, 3:4])+ }))> colnames(MBAs) <- c("N1", "P1",+ "N50", "P50", "N500", "P500")

Now we need to show our results. We are interested in how the relative initialabundances of the two consumers influence the emergence of MBA. Therefore,let’s put the ratio of those two populations (actually the logarithm of the ratio,log[N/P ]9) on an X-axis, and graph the abundances of those two species onthe Y-axis. Finally, we plot side by side the di!erent time points, so we cansee the initial abundances, the transient abundances, and (perhaps) somethingclose to the asymptotic abundances.

> layout(matrix(1:3, nr = 1))> matplot(log10(N.inits[, "N"]/N.inits[,+ "P"]), log10(MBAs[, 1:2] ++ 1), type = "l", main = "Initial Abundances (t=1)",+ col = 2:3, lty = 2:3, lwd = 2,+ ylab = "log(Abundance+1)",+ xlab = "log[N/P]")> legend("right", c("N", "P"), lty = 2:3,+ col = 2:3, bty = "n")> matplot(log10(N.inits[, "N"]/N.inits[,+ "P"]), log10(MBAs[, 3:4] ++ 1), type = "l", main = "At time = 50)",

7 Recall that sapply and related functions “apply” a function (in this case a sim-ulation) to each element of the first argument (in this case each row number ofthe initial abundance matrix).

8 We transpose the output matrix (t()) merely to keep the populations in columns.We also use the hmax argument in lsoda to make sure the ODE solver doesn’ttry to take steps that are too big.

9 logarithms of ratios frequently have much nicer properties than theratios themselves. Compare hist(log(runif(100)/runif(100))) vs.hist(runif(100)/runif(100)).


+ col = 2:3, lty = 2:3, lwd = 2,+ ylab = "log(Abundance+1)",+ xlab = "log[N/P]")> matplot(log10(N.inits[, "N"]/N.inits[,+ "P"]), log10(MBAs[, 5:6] ++ 1), type = "l", main = "At time = 500",+ col = 2:3, lty = 2:3, lwd = 2,+ ylab = "log(Abundance+1)",+ xlab = "log[N/P]")

−3 −2 −1 0 1 2 3

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Initial Abundances (t=1)

log[N/P]

log(Abundance+1)

NP

−3 −2 −1 0 1 2 3

02

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At time = 50)

log[N/P]

log(Abundance+1)

−3 −2 −1 0 1 2 3

02

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At time = 500

log[N/P]log(Abundance+1)

Fig. 8.9: Initial, transient, and near-asymptotic abundances of the intraguild prey,N , and predator, P , of Lotka-Volterra intraguild predation, with di!ering initialabundances.

It is still amazing to me that di!erent initial abundances can have such adramatic e!ect (Fig. 8.10). It is also interesting that they take so long to playout. It all really just makes you wonder about the world we live in.

8.3.4 E!ects of absolute abundance

Now let’s hold relative abundance constant and equal, and vary absolute abun-dance. Recall that in our first explorations, we found di!erent outcomes, de-pending on di!erent total abundances. Now instead of varying N and P inopposite order, we have them vary covary positively.

> logAbs <- seq(2, 7, by = 0.2)> N.abs.inits <- cbind(B = rep(10^9,+ length(logAbs)), N = 10^logAbs,+ P = 10^logAbs)


Now we simulate10 the model, using the same basic approach as above, settingthe time, and hanging on to three di!erent time points.

> t1 <- 1:500> MBA.abs <- t(sapply(1:nrow(N.abs.inits),+ function(i) {+ tmp <- lsoda(N.abs.inits[i,+ ], t1, IGP.ODE, params1,+ hmax = 0.1)+ cbind(tmp[1, 3:4], tmp[50,+ 3:4], tmp[500, 3:4])+ }))> colnames(MBAs) <- c("N1", "P1",+ "N50", "P50", "N500", "P500")

We plot it as above, except that now simply use log10-abundances on thex-axis, rather than the ratio of the di!ering abundances.

> layout(matrix(1:3, nr = 1))> matplot(log10(N.abs.inits[, "N"]),+ log10(MBA.abs[, 1:2] + 1),+ type = "l", main = "Initial Abundances (t=1)",+ col = 2:3, lty = 2:3, lwd = 2,+ ylab = "log(Abundance+1)",+ xlab = expression(log[10]("N")))> legend("right", c("N", "P"), lty = 2:3,+ col = 2:3, bty = "n")> matplot(log10(N.abs.inits[, "N"]),+ log10(MBA.abs[, 3:4] + 1),+ type = "l", main = "At time = 50)",+ col = 2:3, lty = 2:3, lwd = 2,+ ylab = "log(Abundance+1)",+ xlab = expression(log[10]("N")))> matplot(log10(N.abs.inits[, "N"]),+ log10(MBA.abs[, 5:6] + 1),+ type = "l", main = "At time = 500",+ col = 2:3, lty = 2:3, lwd = 2,+ ylab = "log(Abundance+1)",+ xlab = expression(log[10]("N")))

8.3.5 Explanation

Now, . . . we have to explain it! Let’s begin with what we think we know fromLotka-Volterra competition — each species has a bigger e!ect on the others10 Unfortunately ’simulate’ may mean ’integrate,’ as it does here, or any other kind

of made up scenario.


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log[N/P]

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At time = 50)

log[N/P]

log(Abundance+1)

−3 −2 −1 0 1 2 3

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At time = 500

log[N/P]

log(Abundance+1)

Fig. 8.10: Initial, transient, and near-asymptotic abundances of the intraguild prey,N , and predator, P , of Lotka-Volterra intraguild predation, with di!ering initialabundances.

than on itself. How do we apply that here. First let’s look at the per capitadirect e!ects, the parameters for each interaction.

B N PB r ! r"BBB !"BN !"BP

N #NB"BN 0 !"NP

P #PB"BP #PN"NP 0

(8.13)

Then we calculate the values for these and ask if competitors have largere!ects on each other than they do on themselves.

> params1

bpb abp bpn3.2000e-02 1.0000e-08 1.0000e-05

anp mp bnb1.0000e-04 1.0000e+00 4.0000e-02

abn mn r1.0000e-08 1.0000e+00 1.0000e+00

abb3.1623e-10

> with(as.list(params1), {+ rbind(B = c(r - r * abb * 10^9,+ -abn, -abp), N = c(bnb *+ abn, 0, -anp), P = c(bpb *+ abp, bpn * anp, 0))+ })


[,1] [,2] [,3]B 6.8377e-01 -1e-08 -1e-08N 4.0000e-10 0e+00 -1e-04P 3.2000e-10 1e-09 0e+00

> with(as.list(params1), {+ rbind(B = c(r - r * abb * 10^9,+ -abn, -abp), N = c(bnb,+ 0, -anp), P = c(bpb, bpn,+ 0))+ })

[,1] [,2] [,3]B 0.68377 -1e-08 -1e-08N 0.04000 0e+00 -1e-04P 0.03200 1e-05 0e+00

So, from this we are reminded that the per capita direct e!ects on B, the basalresource, by both consumers are the same. N , the IGP prey, however, benefitsmore per capita, and so can attain a higher population size, and thereforecould persist, and also exclude P . Thus it has a larger indirect negative e!ecton P than on itself. P , on the other hand, could have a huge direct negativee!ect on N . To achieve this e!ect, however, P has to have a su"ciently largepopulation size. That is exactly why we get the results we do. If N starts outas relatively abundant, it reduces B and probably excludes P . If, on the otherhand, P is abundant, they can have a large direct negative e!ect on N , andexclude N .

Holt et al. suggest that coexistence is more likely when (i) the IGP preyis the better competitor (as we have above) and (ii) the IGP predator bene-fits substantially from feeding on the IGP prey, that is, when the conversione"ciency of prey into predators, #PN , is relatively large.

Let’s try increasing #PN to test this idea. Let’s focus on the ASS wherethe predator is excluded, when both species start out at low abundances (Fig.8.8, upper right panel). We can focus on the invasion criterion by startingN and B at high abundance and then test whether the predator invade fromlow abundance. We can really ramp up #PN to be 100 times #PB . This mightmake sense if N nutrient value is greater than B. In real food chains this seemsplausible, because the C:N and C:P ratios of body tissue tend to decline asone moves up the food chain [109]; this makes animals more nutritious, insome ways, than plants.

> params2 <- params1> params2["bpn"] <- 0.32

Now we numerically integrate the model.

> t <- 1:100> N.init.1 <- c(B = 10^9, N = 10^5,


+ P = 1)> trial1 <- lsoda(N.init.1, t, IGP.ODE,+ params2)> matplot(t, log10(trial1[, -1] ++ 1), type = "l")

Whoa, dude! Fig. 8.11a reveals the opposite result as Fig. 8.10, but makessense, right? We make the predator benefit a lot more from each prey item,then the predator doesn’t need to be a good competitor, and can persist evenif the IG prey reduces the basal resource level. Our next step is to rein thepredator back in. One way to do this is to reduce the attack rate, so that thepredator has a smaller per capita direct e!ect on the prey. It still benefitsfrom eating prey, but has a harder time catching them. Let’s change "NP

from 10!4 to 10!7 and see what happens.

> params2["anp"] <- 10^-7> trial2 <- lsoda(N.init.1, t, IGP.ODE,+ params2)> matplot(t, log10(trial2[, -1] ++ 1), type = "l")

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log1

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al1[

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, −1]

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(b) High !PN , Low "NP

Fig. 8.11: We can avoid alternate stable states by changing conversion e"cienciesand attack rates. By increasing the direct benefit of prey to the predator, and reducethe attack rate of the predator on the prey, we seem to get coexistence.

Now that we have allowed the predator to benefit more from individualprey, but also made it less likely to attack and kill prey, we get coexistenceregardless of the predator starting at low abundance. Additional explorationis beyond us for the moment, but we have made headway. It turns out thatthis model of intraguild predation yields some very interesting cases, and the


outcomes depend heavily on the productivity of the system (i.e. the carryingcapacity of B). Nonetheless, we have explored the conditions that help facili-tate coexistence of the consumers — the IGP prey is the superior exploitativecompetitor, and the IGP predator benefits substantively from the prey.

Summary

Given a pool of species, we can think of alternate stable states as di!erentspecies combinations which can persistent, and which are not readily invasible(invaders cannot enter at low abundance). This lack of invasibility might comeabout through large direct negative e!ects (high attack rate, or aggression),or through a preempted resource (e.g. light interception, an exclusive substi-tutable resource, or allelopathy). There could also be life history variation,where long lived adults prevent colonization by less competitive juveniles, orjuveniles vulnerable to predation [105].

Problems

8.1. GeneralCompare and contrast the terms“alternate stable states” and“multiple basinsof attraction.”Define each and explain how the terms are similar and how theydi!er.

8.2. Lotka-Volterra competition I(a) Explain what we learn from Fig. 8.2.1 regarding how growth rate, initialabundance and intraspecific density dependence (or carrying capacity) influ-ence outcomes. Specifically, which of these best predicted the final outcomeof competition? Which was worst? Explain.(b) Explain in non-mathematical terms why strong interference allows for pri-ority e!ects.(c) Create a simulation to more rigorously test the conclusions you drew inpart (a) above.

8.3. Resource competition(a) Explain hysteresis.(b) Alter the equation for submerged plants to represent the hypotheticalsituation in which submerged plants get most or all of their resources from thewater column. Explain your rationale, and experiment with some simulations.What would you predict regarding (i) coexistence and (ii) hysteresis? Howcould you test your predictions?

8.4. Intraguild Predation(a) Use Fig. 8.10 to explain how initial abundances influence outcomes. Arethere initial conditions that seem to result in all species coexisting? Are there


things we should do to check this?(b) Explain how high attack rates and low conversion e"ciencies by the toppredator create alternate stable states.

a primer of theoretical population ecology with rstevenmh/stevens_chap8_draft.pdf · m. henry h....

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