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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.

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

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  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

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