Modelling evolution in structured populations involving multiplayer interactions

Transcription

1 Modelling evolution in structured populations involving multiplayer interactions Mark Broom City University London Complex Systems: Modelling, Emergence and Control City University London London June Mark Broom (City University London) City 205 / 36

2 Outline Credits Credits 2 Animal territories 3 Evolutionary grap teory 4 Te model framework 5 Example population structures 6 An evolutionary dynamics and an example game 7 Some results for te fixation probability 8 Discussion and future work Mark Broom (City University London) City / 36

3 Credits Credits Tis work is based upon te papers Broom, M. and Ryctar, J. (202) A general framework for analysing multiplayer games in networks using territorial interactions as a case study Journal of Teoretical Biology , Broom,M., Lafaye, C., Pattni,K. and Ryctar, J. (205) A study of te dynamics of multi-player games on small networks using territorial interactions Journal of Matematical Biology (to appear), and supported by a studentsip funded by te City of London Corporation awarded to Karan Pattni. Mark Broom (City University London) City / 36

4 Outline Animal territories Credits 2 Animal territories 3 Evolutionary grap teory 4 Te model framework 5 Example population structures 6 An evolutionary dynamics and an example game 7 Some results for te fixation probability 8 Discussion and future work Mark Broom (City University London) City / 36

5 Animal territories Animals territorial beaviour Many animals live alone or in distinct groups on a well-defined territory and forage for food almost exclusively witin tat territory. Similarly, males of te species may occupy territories for te purposes of mating. In eiter case, territories will often be defended against rivals and so interactions occur at te boundaries of territories. In tis scenario, we tink of non-overlapping areas wit interaction only at te borders. Often te area tat an animal or group uses for foraging is not exclusive to itself, but can overlap considerably wit te territories of oters. In tis case te more general term ome range is used for te area tat an individual or group utilises. Tere will be parts of te environment claimed by two or more individuals/groups and tere can be interactions between tem, especially over major items of food. Mark Broom (City University London) City / 36

6 Animal territories An example: te territorial beaviour of wild dogs Woodroffe, R Te African Wild Dog: Status Survey and Action Plan. IUCN/SSC Canid Specialist Group, IUCN Publications, Gland Switzerland describes aspects of te territorial beaviour of wild dogs in Africa. Te size of ome ranges varies considerably from site to site, ranging from 500 sq km up to 500 sq km. Packs use smaller areas wen tey are feeding pups at a den. Home range overlap is substantial and varies (from 50% to 80%), see Ginsberg & Macdonald 990. Foxes, Wolves, Jackals and Dogs - An Action Plan for te Conservation of Canids. IUCN/SSC. Gland, Switzerland. Tere are tus parts of territories were tere can be interactions between different dog packs. Te size of te regions of interaction can vary trougout te year. Mark Broom (City University London) City / 36

7 Outline Evolutionary grap teory Credits 2 Animal territories 3 Evolutionary grap teory 4 Te model framework 5 Example population structures 6 An evolutionary dynamics and an example game 7 Some results for te fixation probability 8 Discussion and future work Mark Broom (City University London) City / 36

8 Evolutionary grap teory Evolution on graps Witin te last ten years, models of evolution ave begun to incorporate structured populations using evolutionary grap teory. Let G = (V, E) be a simple, finite, undirected and connected grap, were V is te set of vertices and E is te set of edges. V represents te set of individuals in te population, and E te connections between pairs of individuals. It is usually assumed tat individuals are of two different types, residents (R) and mutants (M). Wen pairs of individuals interact, tey play a game, wit reward a for a mutant against a mutant, b for a mutant against a resident, c for a resident against a mutant and d for a resident against a resident. Individuals play a game against all of teir neigbours, and teir fitness is te average reward from tese contests. Mark Broom (City University London) City / 36

9 Evolutionary grap teory Evolutionary dynamics on graps At te beginning, a vertex is cosen uniformly at random and replaced by a mutant. Subsequently at eac time step, following te Invasion Process dynamics, an individual is selected to reproduce wit probability proportional to its fitness. Te selected individual ten copies itself into a random neigbouring vertex, replacing te individual tere. We follow te set C of vertices occupied by mutant individuals. Te states and V are te absorbing points of te dynamics, and we are particularly interested in te fixation probability, te probability of te end state being V. Mark Broom (City University London) City / 36

10 Evolutionary grap teory An example grap: te star grap Te distinct states of a star grap wit n = 5 leaves. We are typically interested in comparing properties suc as te fixation probability of a grap to tat of te well-mixed population, were every pair of individuals is connected (i.e. E contains all possible edges). Mark Broom (City University London) City / 36

11 Evolutionary grap teory A limitation of evolutionary grap teory One limitation of tis oterwise quite general framework is tat interactions are restricted to pairwise ones, troug te grap edges. Many real animal interactions can involve many players, and teoretical models also describe suc multi-player interactions. We discuss a more general framework of interactions of structured populations focusing on competition between territorial animals. We can embed te results of different evolutionary games witin our structure, as occurs for pairwise games on graps. Grap models ave tree elements: grap, game and dynamics. We tus need a more general mode of interaction, potentially multi-player games and an appropriate dynamics. Mark Broom (City University London) City 205 / 36

12 Outline Te model framework Credits 2 Animal territories 3 Evolutionary grap teory 4 Te model framework 5 Example population structures 6 An evolutionary dynamics and an example game 7 Some results for te fixation probability 8 Discussion and future work Mark Broom (City University London) City / 36

13 Te model framework Te population and its distribution We consider a population of N individuals I,..., I N wo can move to M distinct places P,..., P M. Let X(t) = (X n,m (t)) be a binary N M matrix representing if an individual I n is at place P m ; i.e. {, if I n is at a place P m at time t, X n,m (t) = 0, oterwise. We write P(X(t) = x)(x <t ) = P(X(t) = x X() = x,..., X(t ) = x t ). Let p n,m,t (x <t ) = P(X n,m (t) = )(x <t ) denote te probability of I n being at P m at time t given te istory x <t. Te ome range of an individual I n is defined by P n = {P m : p n,m,t (x <t ) > 0 for some t and istory x <t }. Mark Broom (City University London) City / 36

14 Te model framework Concepts of independence Te population follows a random process, wic can depend upon its entire istory. Tere are simplifications based upon different types of independence. Two important examples are as follows: If a given population distribution is independent of te istory of te process so tat P(X(t) = x)(x <t ) = P(X(t) = x) we call te model istory-independent. We call a istory independent process omogeneous if P(X(t) = x) = P(X = x). If te probability of an individual visiting a place depends only upon te individual and te place, but not upon oter individuals, te istory or te time ten p n,m,t (x <t ) = p n,m n, m, t, x <t. In tis case we simply call te model independent. Mark Broom (City University London) City / 36

15 Te model framework A bipartite grap representation of te independent model P P 2 P m P m P m+ P M P M p 2, p n,m+ p, p 2,2 p,m p n,m p N,M p N,M I I 2 I n I N Te independent model as a bipartite grap. Te weigt between te vertex representing individual I n and patc P m is p n,m. Mark Broom (City University London) City / 36

16 Fitness Te model framework Te reward for individual I n at time t is denoted R(n, x, t, x <t ). If only te current distribution affects te reward, we can use te mean reward as te fitness F n = x P(X = x)r(n, x). Te group G of individuals meeting on P m is G = {I j ; x j,m = }. Let P(X,m = χ G )(x <t ) be te probability of group G meeting on P m at time t. For te independent model we obtain P(X,m = χ G ) = p j,m ( p j,m ). j G Often te reward to an individual will only depend upon te place tat it occupies and te group of individuals on tat place so R(n, x) = R(n, m, χ G ) and F n = j/ G M P(X,m = χ G )(x <t )R(n, m, χ G ). m= G Mark Broom (City University London) City / 36

17 Outline Example population structures Credits 2 Animal territories 3 Evolutionary grap teory 4 Te model framework 5 Example population structures 6 An evolutionary dynamics and an example game 7 Some results for te fixation probability 8 Discussion and future work Mark Broom (City University London) City / 36

18 Example population structures Te territorial interaction model I Consider a scenario were tere are tree individuals I, I 2, I 3 and eac can move freely witin a territory in a sape of a square. Te individual s territories overlap, creating six distinct places P,..., P 6. Assuming te territories are relatively small and tat individuals roam freely and randomly, we may assume tat at any given time, te probability of an individual being on a place witin its own territory is proportional to te area of te place. We tus get an independent model wit (p n,m ) = Mark Broom (City University London) City / 36

19 Example population structures Te territorial interaction model II I 3 I P 6 P 5 P 2 P 3 P 4 P I 2 I 3 I 2 I P 6 P 5 P 4 P 3 P 2 P a) b) Te territory of I is te grey square, te territory of I 2 is te square wit dotted lines, te territory of I 3 is te square wit solid lines. Mark Broom (City University London) City / 36

20 Example population structures Te territorial raider model I Now, consider a special case of te territorial interaction model. Individuals I,..., I N eac occupy teir own place P,..., P N. We consider an example of a star grap wit four individuals. Eac leaf individual as probability of moving to te centre λ. Te centre individual as probability of staying in te centre is µ, going to eac leaf wit equal probability ( µ)/3 oterwise. In particular we consider a specific class wit a single ome fidelity parameter, were µ = /( + 3) and λ = /( + ). We tus ave te following probabilities of movement (p n,m ) = Mark Broom (City University London) City / 36

21 Example population structures Te territorial raider model II P 4 P 3 P µ µ µ P P 2 3 I 4 P 2 P P3 λ λ λ λ λ λ µ I I 2 I 3 a) b) I 4 I 3 I 2 I c) a) Individual I n lives in place P n but can raid neigbouring places. Te territory of I is te wole triangle, te territory of I 2 is te rombus encompassed by full lines etc; b) representation as a general independent model; c) visualization as multi-player interactions on a grap. Mark Broom (City University London) City / 36

22 Example population structures Te territorial raider model III Te population structures and movement probabilities for small graps on 3 and 4 vertices. (a) 3 vertex line, (b) triangle, (c) 4 vertex complete grap, (d) 4 vertex circle", (e) 4-vertex star, (f) diamond (g) 4 vertex line, () paw. Mark Broom (City University London) City / 36

23 Example population structures Te territorial raider model IV Te transition graps for small graps on 3 and 4 vertices. Mark Broom (City University London) City / 36

24 An evolutionary dynamics and an example game Outline Credits 2 Animal territories 3 Evolutionary grap teory 4 Te model framework 5 Example population structures 6 An evolutionary dynamics and an example game 7 Some results for te fixation probability 8 Discussion and future work Mark Broom (City University London) City / 36

25 An evolutionary dynamics and an example game An evolutionary dynamics I Let b i denote te probability an individual I i is selected for reproduction were, b i = F i / k F k. Let d ij, for i j, denote te probability tat I j is replaced by a copy of I i given I i is selected for reproduction. We calculate d ij by considering all possible places P m and all possible groups G {, 2,..., N} involving bot individuals i and j; weigted by χ(m, G), te probability of te group G meeting at place P m, and by a factor ( G ) representing te fact tat in a group G, an individual I i could replace any one of G oter individuals. d ij = N χ(m, G) = k G m= G:i,j G χ(m, G) G, were p km ( p k m). k G Mark Broom (City University London) City / 36

26 An evolutionary dynamics and an example game An evolutionary dynamics II Letting P SS denote te transition probability from state S to state S in te dynamic process of our game, for S S we ave b i d ij ; if S = S \ {j} for some j S i S P SS = b i d ij ; if S = S {j} for some j S i S 0; oterwise and we set P SS = S S P SS. Mark Broom (City University London) City / 36

27 An evolutionary dynamics and an example game An evolutionary dynamics III Te temperature of te I-vertex I j is T j = i j d ij. Let ρ A S be te probability tat A fixates from state S, were ρ A S = P SS ρ A S S {,2,...,N} wit boundary conditions ρ A = 0, ρa V =. Te mean fixation probability of A, ρ A, is defined as ρ A = i T i ρ A {i} T. j j Mark Broom (City University London) City / 36

28 An evolutionary dynamics and an example game An example game We consider a multi-player game wit strategies Hawk and Dove, competing for a single reward, value V, were eac individual as a background reward R irrespective of teir strategy. If all individuals in a figting group are Doves, tey split te reward, so eac receives te reward divided by te number in te group. If tere are any Hawks, all te Doves flee and get 0, all te Hawks figt and one of tem receives te reward, and all of te oters receive a cost C. Tus if we denote R D a,b (RH a,b ) te reward for a Hawk (Dove) witin a group wit a Hawks and b Doves (including itself), we get R H a,b = R + V (a )C, R D a,b = a { R; if a > 0 R + V b ; if a = 0. Mark Broom (City University London) City / 36

29 Outline Some results for te fixation probability Credits 2 Animal territories 3 Evolutionary grap teory 4 Te model framework 5 Example population structures 6 An evolutionary dynamics and an example game 7 Some results for te fixation probability 8 Discussion and future work Mark Broom (City University London) City / 36

30 Some results for te fixation probability Hawk fixation probability in a Dove population Te fixation probabilities of a single Hawk in a population of Doves for small graps on 3 and 4 vertices. Average fixation probability Line(3), Hawks in Hawk Dove game v= 2 v= v= 0.5 v= 0.25 v= 0.05 Average fixation probability Triangle, Hawks in Hawk Dove game v= 2 v= v= 0.5 v= 0.25 v= 0.05 Average fixation probability Complete(4), Hawks in Hawk Dove game v= 2 v= v= 0.5 v= 0.25 v= 0.05 Average fixation probability Square, Hawks in Hawk Dove game v= 2 v= v= 0.5 v= 0.25 v= Average fixation probability Star(4), Hawks in Hawk Dove game v= 2 v= v= 0.5 v= 0.25 v= 0.05 Average fixation probability Diamond, Hawks in Hawk Dove game v= 2 v= v= 0.5 v= 0.25 v= 0.05 Average fixation probability Line(4), Hawks in Hawk Dove game v= 2 v= v= 0.5 v= 0.25 v= 0.05 Average fixation probability Paw, Hawks in Hawk Dove game v= 2 v= v= 0.5 v= 0.25 v= Mark Broom (City University London) City / 36

31 Some results for te fixation probability Dove fixation probability in a Hawk population Te fixation probabilities of a single Dove in a population of Hawks for small graps on 3 and 4 vertices Line(3), Doves in Hawk Dove game 0.34 Triangle, Doves in Hawk Dove game Complete(4), Doves in Hawk Dove game Square, Doves in Hawk Dove game Average fixation probability v= 2 v= v= v= 0.25 v= Average fixation probability v= 2 v= v= v= 0.25 v= Average fixation probability v= 2 v= v= 0.5 v= 0.25 v= Average fixation probability v= 2 v= v= 0.5 v= 0.25 v= Star(4), Doves in Hawk Dove game Diamond, Doves in Hawk Dove game Line(4), Doves in Hawk Dove game Paw, Doves in Hawk Dove game Average fixation probability v= 2 v= v= 0.5 v= 0.25 v= Average fixation probability v= 2 v= v= 0.5 v= 0.25 v= Average fixation probability v= 2 v= v= 0.5 v= 0.25 v= Average fixation probability v= 2 v= v= 0.5 v= 0.25 v= Mark Broom (City University London) City / 36

32 Outline Discussion and future work Credits 2 Animal territories 3 Evolutionary grap teory 4 Te model framework 5 Example population structures 6 An evolutionary dynamics and an example game 7 Some results for te fixation probability 8 Discussion and future work Mark Broom (City University London) City / 36

33 Discussion I Discussion and future work We ave developed a new framework for modelling game teoretical interactions in a structured population. Te framework incorporates tree key components: population structure; evolutionary dynamics; evolutionary game. A useful area of application of our model is in animal territorial beaviour. Different species can exibit different types of territoriality, and a flexible system of modelling tis is required. Mark Broom (City University London) City / 36

34 Discussion II Discussion and future work Evolutionary grap teory as made, and continues to make, important contributions to te understanding of te effect of population structure. Our framework as some advantages over standard evolutionary grap teory. For example, multi-player games can be explicitly modelled in our structure. In addition, tere is a natural (and more logical) way of converting aggregated game payoffs into fitness. Mark Broom (City University London) City / 36

35 Discussion III Discussion and future work Following a recently accepted paper, we ave considered an example wic incorporates all tree key aspects of our framework. In particular we ave developed a birt-deat dynamics and a natural way to find te fixation probability of a rare mutant for any population structure. We ave seen tat key features of te structure, including te temperature and te mean group size, ave a strong influence on te fixation probability. For example, ig temperature amplifies te influences of te key game parameters like te reward, increasing te fixation probabilities of te fitter individuals, and decreasing te fixation probabilities of te weaker individuals. Mark Broom (City University London) City / 36

36 Discussion IV Discussion and future work An important next step in tis work is to more fully incorporate evolutionary dynamics in te new framework. In evolutionary grap teory, tere are a number of dynamics reflecting different biological scenarios. In particular we need to develop deat-birt dynamics, as well as alternative birt-deat dynamics. Anoter aspect tat needs development is te analysis of more realistic populations. Tis requires larger populations, and also populations tat ave differing numbers of places and individuals, i.e. wic are not grap-like. Te above work is ongoing. We note tat tis researc is still in its relatively early stages. Mark Broom (City University London) City / 36