1 / 27

Theoretical Modelling in Biology Pt. II: Numerical Models for Multispecies Models

This course provides a short introduction to programming, data representation, operations on data, decision making, and iteration. It also covers numerical integration, age-dependent reproduction and mortality rate in individuals, and the interactions between multiple species within a community.

frisk
Download Presentation

Theoretical Modelling in Biology Pt. II: Numerical Models for Multispecies Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Theoretical Modelling in Biology (G0G41A )Pt II. Numerical ModelsII. multispecies models Joost Vanoverbeke Dept. of Biology, K.U.Leuven

  2. (Very) short introduction to programming • Data representation • Operations on data: functions • Decision making: conditional expressions • Iteration

  3. Data representation • Basic data types: • Integer ..., -1, 0, 1, 2, 3, ... • Real = floating point 1.5, 3.33333333, 0.0004, ... • Character ‘a’, ‘b’, ‘c’, ... ‘1’, ‘2’, ‘.’, ‘&’, ... • Bool true – false; 1 – 0

  4. Data representation • Variables & constants int x = 5; x => 5 float y = 3.2; char c = ‘d’; const int z 6; x = 7; OK z = 3; NO

  5. Data representation ‘H’ 1 ‘e’ 5 ‘l’ 3 ‘l’ 34 7 ‘o’ 18 ‘ ’ ‘w’ 12 ... ‘o’ ‘r’ ‘l’ ‘d’ • Compound data types • Array, Vector; (List) A = A[2] => 5; A[6] => 18 • String = array of characters “Hello world” =

  6. Operations on data • function sum(x, y) = x + y; a = 2; b = 3; c = sum(a, b); c => 5 • function double_sum(x, y) = sum(x, y) + sum(x, y); double_sum(a, b) => 10

  7. Conditional expression • Boolean expression: result = true or false 3 < 5 => true 4 >= 7 => false • Conditional expression: if then else if <boolean expression> = true then do <expression 1> else do <expression 2> • a = 5; b = 7; if a < 4 then sum(a, b) else double_sum(a, b) => 24

  8. Repeat a give sequence of programming instructions for a number of times Predefined number of times: for x between 1 and 10 do { instructions...; x=x+1; } No predefined number of times: while x < 10 do { instructions...; x = a_function(x) } Numerical integration = loop over discrete time intervals ∆t for time between 1 and 100 do { N = dN/dt * ∆t; time = time + ∆t; } Iteration & loops

  9. Exercise 3: individual based model • Extend the given individual based model (‘indivpop1.sml’) to include age dependent reproduction rate and mortality rate • Age of an individual: • can be calculated as time(1) – init_time(1); both time() and init_time() are build-in functions in simile • Reproduction: • Individuals will only reproduce when their age is equal or larger than 10 • This can be programmed by using an ‘if ... then ... else ...’ statement. • Mortality: • Make mortality linearly dependent on age: mortality = d*age • Plot the population size over time • Calculate the average age of the individuals and plot over time

  10. Multispecies models • Modelling the interactions between multiple species within a community

  11. Two species competitionN1(- -)N2 Stable coexistence N2 K1/ α12 > K2 N1 K1 > K2/ α21 • Stable coexistence if K1 < K2/α21 K2 < K1/α12 • Competition is weak: intraspecific competition > interspecific competition

  12. Interactions between more than two species with α11 = 1

  13. Stable coexistence Prey only P P N N < K K < Lotka Volterra Predator PreyN1(- +)N2 h = prey capture rate β = conversion efficiency Stable coexistence if K > m/(βh) Carrying capacity > output/input budget of predator

  14. Reformulate : Prey (N1) : intrapspecific response -r/K = a11 Prey : interspecific response -h = a12 Predator (N2) : intrinsic rate of increase b – m with b = 0 => -m = r2 Predator : intraspecific response 0 = a22 Predator : interspecific response βh = a21 Lotka Volterra Predator PreyN1(- +)N2 =>

  15. Multispecies models Competition : -riαij/Ki = aij All Lotka-Volterra interactions can be written as: aij(- -)aji : competition aij(+ +)aji : mutualism aij(- +)aji : predator – prey aij(0 +)aji : commensalism

  16. Community model: Focus on all possible interactions between species (- -), (- +), (+ +), (+ 0) Food web model: Focus on predator-prey interactions (- +) 8 8 9 9 4 4 5 5 7 7 6 6 1 1 3 3 2 2 Community and food web models

  17. Characteristics of multispecies models • Size of the community or food web • S : Number of species • Number of ‘trophic’ species: collective term for all the species having a common set of predators and prey • Complexity • L : total number of links between species in food web • Relative to S: • Linkage density : L/S • Connectance : L/S(S-1) With S(S-1) = total possible number of interactions between S species

  18. 8 8 9 9 4 4 5 5 7 7 6 6 1 1 3 3 2 2 Characteristics of multispecies models • Type of species & Trophic levels • Top level: species have no predators • Intermediate level(s): species have both predators and prey • Basal level: species have no prey • Obtain their resources directly from the environment • Length of food chains • Food chain: set of (predator prey) links along a particular path starting from a basal species and endig at a top species • Mean chain length

  19. Stability analysis of multispecies models • Equilibrium densities of the species? • Dependence on number of species S? • Dependence on number of links L? • Dependence on type of interactions? (- -), (- +), (+ +), (+ 0)

  20. Equilibrium densities • Set of coupled equations: • Solve for equilibrium solutions for all Ni • Are equilibrium solutions stable or unstable?

  21. Stability analysis • Analytical solution: • Jacobian matrix = matrix with partial derivatives of growth rate functions dNi/dt on each Nj • Solve for the eigenvalues of the Jacobian matrix for each equilibrium solution • Parameter settings (ri, aij) for which all eigenvalues have negative real parts give stable equilibrium solutions • For number of species S >= 5 general analytical solutions are impossible • Other approaches for stability analysis needed

  22. Stability analysis • Approach 1 • Numerical simulation of the trajectories of the species densities Ni over time • Analyse the simulation results for different parameter settings • Approach 2 • Numerical solutions of the Jacobian matrix for particular parameter settings • Find solutions for different parameter settings • Determine parameter settings for which equilibria are stable

  23. Stability analysis • Approach 3: • Derive qualitative rules • Given the relative strength of intraspecific (aii) versus interspecific (aij) interactions, what is the likelihood of community stability? • Given the signs of the elements aij, what is the likelihood of community stability? • Some qualitative rules: • Increasing the number of species S will likely decrease stability of community and food web models • Increasing connectance L/S(S-1) will also likely decrease stability • Strong intraspecific (aii) and weak interspecific (aij) interactions will increase community stability • Predator-prey interactions are more likely to be stable than other types of interactions => different trophic levels increase probability of stable food webs

  24. Exercise 4: food webs • Given a community with one predator and two competing prey species: • With m = 0.3; β1 = 0.2; h1 = 0.1; β2 = 0.1; h2 = 0.1 • And r1 = 0.5; K1 = 100; α12 = 0.7; r2 = 0.4; K2 = 120; α21 = 0.8 • Reformulate the model in terms of aij and calculate their value • Open the simile file ‘foodweb.sml’ and change the model to the reformulated model in terms of aij values • Start the model with initial densities P = 5, N1 = 10, N2 = 10 and plot the species densities over time • Is there stable coexistence of all three species?

  25. Exercise 4: food webs • dP/dt = P(rP + ap1N1 + ap2 N2) • dN1/dt = N1(r1 + a11N1 + a12N2 + a1PP) • dN2/dt = N2(r2 + a22N2 + a21N1 + a2PP) • ap1 = β1h1 • …

  26. Example multispecies competition • ‘MultiComp1.sml’ • Multispecies competition model for arbitrary number of competing species • For all species: intraspecific competition (aij for i = j) equals -0.05 • Interspecific competition strength (aij for i ≠ j) is determined at random during initialization of the model all aij range between -0.05 and 0 • Such a model can be used for example to analyse effect of community size, connectance and relative importance of intra- versus interspecific interactions on community stability

  27. Exercise 5: multispecies competition • Open the simile file ‘MultiComp1.sml’ • Adapt the simile model to represent the following multispecies competition model: • Kiare randomly created during initialization of the model andrange between 50 and 100 (use the build in rand_const() function) • αij = 1 for i = j ; for i ≠ j , αijare randomly created during initialization of the model andrange between 0.7 and 1.1 • Start the model with initial densities Ni = 10

More Related