Modeling Nature February 2009 1 Modeling Nature LECTURE

Modeling Nature February 2009 1

Modeling Nature LECTURE 2: Predator-prey models * and more general information … 2

Task descriptions and required readings • Each lecture has two associated tasks (a, b) • The task descriptions can be found on ELEUM • Each task has a number of required readings on the ELEUM website • Additional readings and Web pointers are also available (not mandatory, but useful) 3

Task Descriptions • Updated task descriptions of are made available through ELEUM on a weekly basis • Note that the task descriptions in the course manual (syllabus) become overruled by these new descriptions !!! 4

Student research project A TEAM consists of 2 to 3 students A 2 -3 PAGE PROPOSAL is submitted ultimately on Monday 2 March to the tutor of the team’s TG. PRESENTATION: +/-10 minutes each presentation in the week of 23 March in the TGs. REPORT: a short paper (2500 words) on the subject. THE GRADE is for the TEAM and thus for all students in the TEAM. It consists of 50% for the report and 50% for the presentation. 5

Complete time table • • 02/2 - Lecture 1 : Models, Growth & Decay 09/2 - Lecture 2 : Predator-Prey Systems 16/2 - Lecture 3 : Network Models 02/3 - Lecture 4 : Chaos and Fractals; first draft report and presentation – feedback from tutors • 09/3 - Lecture 5 : Percolation and Phase Transitions • 16/3 - Lecture 6 : Self-Organization and Collective Phenomena • 23/3 - Week 7 : Student presentations on chosen topic; report • 30/3 - Week 8 : Final exam 6

Final mark of the course The final mark of the course consists of 40% of the project and 60% of the written exam. Only students with sufficient attendance may attend the exam and present his/her project, when only 1 TG lacks for a valid pass, the student receives an additional task, otherwise the students fails the course. 7

General Information More questions? Ask after the Lecture or your personal tutor 8

Lecture 2: PREDATOR-PREY SYSTEMS 9

Overview • From one to two equations • Volterra’s model of predator-prey (PP) systems • Why are PP models useful? • Examples from nature • Relation to future tasks 10

Recall the Logistic Model Large P slows down P Logistic model a. k. a. the Verhulst model • Pn is the fraction of the maximum population size 1 • is a parameter that describes the strength of the coupling 11

Interacting quantities • The logistic model describes the dynamics (change) of a single quantity interacting with itself • We now move to models describing two (or more) interacting quantities 12

Fish statistics • Vito Volterra (1860 -1940): a famous Italian mathematician • Father of Humberto D'Ancona, a biologist studying the populations of various species of fish in the Adriatic Sea • The numbers of species sold on the fish markets of three ports: Fiume, Trieste, and Venice. 13

percentages of predator species (sharks, skates, rays, . . ) 14

Volterra’s model • Two (simplifying) assumptions – The predator species is totally dependent on the prey species as its only food supply – The prey species has an unlimited food supply and no threat to its growth other than the specific predator prey 15

Lotka–Volterra equation : predator prey The Lotka–Volterra equations are a pair of equations used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey. They were proposed independently by Alfred J. Lotka in 1925 and Vito Volterra in 1926. 16

Lotka–Volterra equation : Two species #1: population size: x species #2: population size: y 17

Lotka–Volterra equation : Remember Verhulst-equation: Predator (x) and prey (y) model: xn+1 = xn(α – βyn) yn+1 = yn(γ – δxn) : y is the limitation for x : x is the limitation for y 18

Lotka–Volterra equation : Equivalent formulation: rate of change: dx/dt = in/de-crease per unit time (e. g. 2000 hares per year) 19

Behaviour of the Volterra’s model Oscillatory behaviour Limit cycle 20

Effect of changing the parameters (1) Behaviour is qualitatively the same. Only the amplitude changes. 21

Effect of changing the parameters (2) Behaviour is qualitatively different. A fixed point instead of a limit cycle. 22

Different modes… 23

Predator-prey interaction in vivo Huffaker (1958) reared two species of mites to demonstrate coupled oscillations of predator and prey densities in the laboratory. He used Typhlodromus occidentalis as the predator and the 24 six-spotted mite (Eotetranychus sexmaculatus) as the prey

Online simulation of PP model • http: //www. xjtek. com/models/? archive=ecosystem_dynamics/predat or_prey/model. jar, xjanylogic 6 engine. jar&root=predator_prey. Simulat ion$Applet&width=800&height=650&version=6 25

Why are PP models useful? • They model the simplest interaction among two systems and describe natural patterns • Repetitive growth-decay patterns, e. g. , – World population growth – Diseases –… Exponential growth Limited growth Exponential decay Oscillation 26 time

Lynx and hares Very few "pure" predator-prey interactions have been observed in nature, but there is a classical set of data on a pair of interacting populations that come close: the Canadian lynx and snowshoe hare pelt-trading records of the Hudson Bay Company over almost a century. 27

Lynx and hares 28

The Hudson Bay data give us a reasonable picture of predator-prey interaction over an extended period of time. The dominant feature of this picture is the oscillating 29 behavior of both populations

1. what is the period of oscillation of the lynx population? 2. what is the period of oscillation of the hare population? 3. do the peaks of the predator population match or slightly precede or slightly lag those of the prey population? 30

“equilibrium” states • Complex systems are assumed to converge towards an equilibrium state. Equilibrium state: two (or more) opposite processes take place at equal rates 31 unstable

Adaptations 32

Evolutionary arms race 33

This is the basis for evolution 34

This is the basis for evolution 35

More complicated interactions [1] • Clinton established the Giant Sequoia National Monument to protect the forest from culling, logging and clearing. – But many believe that Clinton’s measures added fuel to the fires. – Tree-thinning is required to prevent large fires. – Fires are required to clear land to promote new growth. 36

Sequoias 37

Predator versus Prey? Fire is dangerous when caused by surrounding bushes Fire is needed to clean area and to open the seeds of the Sequoia • Fire acts as “prey” because it is needed for growth • Fire acts as “predator” because it may set the tree on fire • Tree acts as “prey” for the predator • If trees die out, the 38 predator dies out too

More complicated interactions [2] Predators, Preys and Hurricanes 39

More complicated interactions [3] Biodiversity “Human alteration of the global environment has triggered the sixth major extinction event in the history of life and caused widespread changes in the global distribution of organisms. These changes in biodiversity alter ecosystem processes and change the resilience of ecosystems to environmental change. This has profound consequences for services that humans derive from ecosystems. The large ecological and societal consequences of changing biodiversity should be minimized to preserve options for future solutions to global environmental problems. ” F. Stuart Chapin III et al. (2000)40

The role of biodiversity in global change 41

Consequences of reduced biodiversity ". . . decreasing biodiversity will tend to increase the overall mean interaction strength, on average, and thus increase the probability that ecosystems undergo destabilizing dynamics and collapses. " Kevin Shear Mc. Cann (2000) 42

Relation to the. Tasks 43

Predation, competition, and interaction 44

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END of LECTURE 2 48