Modelling evolution in structured populations involving …€¦ · 4 The model framework 5 Example...

Modelling evolution in structured populationsinvolving multiplayer interactions

Mark Broom

City University London

Game Theoretical Modelling of Evolution in Structured PopulationsNIMBioSKnoxville

25-27 April 2016

Mark Broom (City University London) NIMBioS 2016 1 / 39

Credits

Outline

1 Credits

2 Animal territories

3 Evolutionary graph theory

4 The model framework

5 Example population structures

6 An evolutionary dynamics and an example game

7 Some results for the fixation probability

8 Discussion and future work


Credits

Credits

This work is based upon the papers

Broom, M. and Rychtár, J. (2012) A general framework for analysingmultiplayer games in networks using territorial interactions as a casestudy Journal of Theoretical Biology 302 70-80,

Broom,M., Lafaye, C., Pattni,K. and Rychtár, J. (2015) A study of thedynamics of multi-player games on small networks using territorialinteractions Journal of Mathematical Biology 71 1551-1574,

and supported by a studentship funded by the City of London Corporationawarded to Karan Pattni.


Animal territories

Outline

1 Credits









Animal territories

Animals’ territorial behaviour

Many animals live alone or in distinct groups on a well-defined territoryand forage for food almost exclusively within that territory. Similarly,males of the species may occupy territories for the purposes of mating.

In either case, territories will often be defended against rivals and sointeractions occur at the boundaries of territories. In this scenario, wethink of non-overlapping areas with interaction only at the borders.

Often the area that an animal or group uses for foraging is not exclusiveto itself, but can overlap considerably with the territories of others.

In this case the more general term home range is used for the area that anindividual or group utilises.

There will be parts of the environment claimed by two or moreindividuals/groups and there can be interactions between them,especially over major items of food.


Animal territories

An example: the territorial behaviour of wild dogs

Woodroffe, R. 1997. The African Wild Dog: Status Survey and Action Plan.IUCN/SSC Canid Specialist Group, IUCN Publications, Gland Switzerlanddescribes aspects of the territorial behaviour of wild dogs in Africa.

The size of home ranges varies considerably from site to site, rangingfrom 500 sq km up to 1500 sq km.

Packs use smaller areas when they are feeding pups at a den.

Home range overlap is substantial and varies (from 50% to 80%), seeGinsberg & Macdonald 1990. Foxes, Wolves, Jackals and Dogs - An ActionPlan for the Conservation of Canids. IUCN/SSC. Gland, Switzerland.

There are thus parts of territories where there can be interactionsbetween different dog packs.

The size of the regions of interaction can vary throughout the year.


Evolutionary graph theory

Outline

1 Credits










Evolution on graphs

Within the last ten years, models of evolution have begun to incorporatestructured populations using evolutionary graph theory.

Let G = (V,E) be a simple, finite, undirected and connected graph,where V is the set of vertices and E is the set of edges.

V represents the set of individuals in the population, and E theconnections between pairs of individuals.

It is usually assumed that individuals are of two different types, residents(R) and mutants (M).

When pairs of individuals interact, they play a game, with reward a for amutant against a mutant, b for a mutant against a resident, c for a residentagainst a mutant and d for a resident against a resident.

Individuals play a game against all of their neighbours, and their fitnessis the average reward from these contests.



Evolutionary dynamics on graphs

At the beginning, a vertex is chosen uniformly at random and replacedby a mutant.

Subsequently at each time step, following the Invasion Processdynamics, an individual is selected to reproduce with probabilityproportional to its fitness.

The selected individual then copies itself into a random neighbouringvertex, replacing the individual there.

We follow the set C of vertices occupied by mutant individuals.

The states ∅ and V are the absorbing points of the dynamics, and we areparticularly interested in the fixation probability, the probability of theend state being V .



An example graph: the star graph

The distinct states of a star graph with n = 5 leaves.We are typically interested in comparing properties such as the fixationprobability of a graph to that of the well-mixed population, where every pairof individuals is connected (i.e. E contains all possible edges).



A limitation of evolutionary graph theory

One limitation of this otherwise quite general framework is thatinteractions are restricted to pairwise ones, through the graph edges.

Many real animal interactions can involve many players, and theoreticalmodels also describe such multi-player interactions.

We discuss a more general framework of interactions of structuredpopulations focusing on competition between territorial animals.

We can embed the results of different evolutionary games within ourstructure, as occurs for pairwise games on graphs.

Graph models have three elements: graph, game and dynamics. We thusneed a more general mode of interaction, potentially multi-player gamesand an appropriate dynamics.


The model framework

Outline

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The model framework

The population and its distribution

We consider a population of N individuals I1, . . . , IN who can move to Mdistinct places P1, . . . ,PM.

Let X(t) = (Xn,m(t)) be a binary N ×M matrix representing if anindividual In is at place Pm; i.e.

Xn,m(t) =

{1, if In is at a place Pm at time t,0, otherwise.

We writeP(X(t) = x)(x<t) = P(X(t) = x|X(1) = x1, . . . ,X(t − 1) = xt−1).

Let pn,m,t(x<t) = P(Xn,m(t) = 1)(x<t) denote the probability of In beingat Pm at time t given the history x<t .

The home range of an individual In is defined byPn = {Pm : pn,m,t(x<t) > 0 for some t and history x<t}.


The model framework

Concepts of independence

The population follows a random process, which can depend upon itsentire history. There are simplifications based upon different types ofindependence. Two important examples are as follows:

If a given population distribution is independent of the history of theprocess so that

P(X(t) = x)(x<t) = P(X(t) = x)

we call the model history-independent. We call a history independentprocess homogeneous if P(X(t) = x) = P(X = x).

If the probability of an individual visiting a place depends only upon theindividual and the place, but not upon other individuals, the history or thetime then

pn,m,t(x<t) = pn,m ∀n,m, t, x<t.

In this case we simply call the model independent.


The model framework

A bipartite graph representation of the independentmodel

I1 I2 In

P1 P2 Pm

IN

Pm!1 Pm+1 PM!1 PM

pn,m

pn,m+1

pN,M!1

pN,Mp1,1

p1,m!1

p2,2

p2,1

The independent model as a bipartite graph. The weight between the vertexrepresenting individual In and patch Pm is pn,m.


The model framework

Fitness

The reward for individual In at time t is denoted R(n, x, t, x<t).If only the current distribution affects the reward, we can use the meanreward as the fitness Fn =

∑x P(X = x)R(n, x).

The group G of individuals meeting on Pm is G = {Ij; xj,m = 1}.Let P(X◦,m = χG)(x<t) be the probability of group G meeting on Pm attime t. For the independent model we obtain

P(X◦,m = χG) =∏j∈G

pj,m

∏j/∈G

(1− pj,m).

Often the reward to an individual will only depend upon the place that itoccupies and the group of individuals on that place soR(n, x) = R(n,m, χG) and

Fn =

M∑m=1

∑G

P(X◦,m = χG)(x<t)R(n,m, χG).


Example population structures

Outline

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The territorial interaction model I

Consider a scenario where there are three individuals I1, I2, I3 and eachcan move freely within a territory in a shape of a square.

The individual’s territories overlap, creating six distinct placesP1, . . . ,P6.

Assuming the territories are relatively small and that individuals roamfreely and randomly, we may assume that at any given time, theprobability of an individual being on a place within its own territory isproportional to the area of the place.

We thus get an independent model with

(pn,m) =

12

14

14 0 0 0

0 0 14

12

14 0

0 14

14 0 1

414

.



The territorial interaction model II

P1

P2 P3P4

P5P6

I1

I2I3

I1

I2

I3

P1

P2

P3

P4

P5

P6

a) b)

12

12

14 1

4141

4

14

14 1

4

The territory of I1 is the grey square, the territory of I2 is the square withdotted lines, the territory of I3 is the square with solid lines.



The boundary interactions model I

Consider a scenario where there are four individuals I1, I2, I3, I4 and eachcan move freely within an area in a shape of a regular hexagon.

The interaction between individuals can only occur at the boundaries.

Interactions can only be pairwise with the pn,m given below

(pn,{i,j}) =

23

13 0 0 0

24 0 1

414 0

0 13

13 0 1

3

0 0 0 12

12

We consider a graph with I1, . . . , IN as individuals on vertices, and placesas edges, as shown in the next figure.



The boundary interactions model II

I2

I1

I3

P{1,2}

I2I1

I3

I1

I2

I3

23

13

a) b) c)

P{2,3}

P{1,3}

P{1,2}

P{2,3}P{1,3}

P{1,3}

P{2,3}

P{1,2}

I4

P{2,4}

P{3,4}I4

P{2,4}

P{3,4}12

12

24

14

14

13

13

13

I4P{3,4}

P{2,4}

Boundary interaction model. a) Individuals guarding their areas; b)representation as a general independent model; c) visualizationas pairwise interactions on a graph.



The boundary interactions model III

Let A = (Ai,j) be the adjacency matrix of the graph, i.e. Ai,j = 1 if thereis an edge between Ii and Ij and Ai,j = 0 otherwise.

Suppose that In has degree dn =∑

j An,j, so its possible groups are either{In} or {In, In′} for each of the dn individuals In′ such that An,n′ = 1.

Assuming that the reward for a given individual being alone does notdepend upon which boundary of its territory it is at, we can obtain

Rn =∑

n′

An,n′

dn

(1

dn′fn,{n,n′} + (1− 1

d′n)fn,{n}

)= fn,{n} +

∑n′

An,n′1dn

1dn′

[fn,{n,n′} − fn,{n}

]where fn,{n} is the payoff when being alone, fn,{n,n′} is the payoff whenbeing with an individual n′.



The territorial raider model I

Now, consider a special case of the territorial interaction model.Individuals I1, . . . , IN each occupy their own place P1, . . . ,PN .We consider an example of a star graph with four individuals.Each leaf individual has probability of moving to the centre λ. Thecentre individual has probability of staying in the centre is µ, going toeach leaf with equal probability (1− µ)/3 otherwise.In particular we consider a specific class with a single home fidelityparameter h, where µ = h/(h + 3) and λ = 1/(h + 1).We thus have the following probabilities of movement

(pn,m) =

hh+3

1h+3

1h+3

1h+3

1h+1

hh+1 0 0

1h+1 0 h

h+1 01

h+1 0 0 hh+1

.



The territorial raider model II

P1

P2 P3

P4

P3 P1P2P4

I1I2I3I4

!

µ

! !

1!!1!!

1!!

1!µ3

1!µ3

1!µ3

a) b)

I1

I2 I3

I4

c)

a) Individual In lives in place Pn but can raid neighbouring places. Theterritory of I1 is the whole triangle, the territory of I2 is the rhombusencompassed by full lines etc; b) representation as a general independentmodel; c) visualization as multi-player interactions on a graph.



The territorial raider model III

The population structures and movement probabilities for small graphs on 3and 4 vertices. (a) 3 vertex line, (b) triangle, (c) 4 vertex complete graph, (d) 4vertex “circle", (e) 4-vertex star, (f) diamond (g) 4 vertex line, (h) paw.



The territorial raider model IV

The transition graphs for small graphs on 3 and 4 vertices.


An evolutionary dynamics and an example game

Outline

1 Credits










An evolutionary dynamics I

Let bi denote the probability an individual Ii is selected for reproductionwhere, bi = Fi/

∑k Fk.

Let dij, for i 6= j, denote the probability that Ij is replaced by a copy of Ii

given Ii is selected for reproduction.We calculate dij by considering all possible places Pm and all possiblegroups G ⊂ {1, 2, . . . ,N} involving both individuals i and j; weighted byχ(m,G), the probability of the group G meeting at place Pm, and by afactor (|G| − 1)−1 representing the fact that in a group G, an individual Ii

could replace any one of |G| − 1 other individuals.

dij =

N∑m=1

∑G:i,j∈G

χ(m,G)

|G| − 1, where

χ(m,G) =∏k∈G

pkm

∏k′ 6∈G

(1− pk′m).



An evolutionary dynamics II

Letting PSS′ denote the transition probability from state S to state S′ in thedynamic process of our game, for S 6= S′ we have

PSS′ =

∑i6∈S

bidij; if S′ = S \ {j} for some j ∈ S∑i∈S

bidij; if S′ = S ∪ {j} for some j 6∈ S

0; otherwise

and we setPSS = 1−

∑S′ 6=S

PSS′ .



An evolutionary dynamics III

The temperature of the I-vertex Ij is

Tj =∑i6=j

dij.

Let ρAS be the probability that A fixates from state S, where

ρAS =

∑S′⊂{1,2,...,N}

PSS′ρAS′

with boundary conditions ρA∅ = 0, ρA

V = 1.

The mean fixation probability of A, ρA, is defined as

ρA =∑

i

Ti∑j

TjρA{i}.



An example game

We consider a multi-player game with strategies Hawk and Dove,competing for a single reward, value V , where each individual has a“background” reward R irrespective of their strategy.

If all individuals in a fighting group are Doves, they split the reward, soeach receives the reward divided by the number in the group.

If there are any Hawks, all the Doves flee and get 0, all the Hawks fightand one of them receives the reward, and all of the others receive a costC.

Thus if we denote RDa,b(R

Ha,b) the reward for a Hawk (Dove) within a

group with a Hawks and b Doves (including itself), we get

RHa,b = R +

V − (a− 1)Ca

, RDa,b =

{R; if a > 0R + V

b ; if a = 0.


Some results for the fixation probability

Outline

1 Credits










Hawk fixation probability in a Dove population

The fixation probabilities of a single Hawk in a population of Doves for smallgraphs on 3 and 4 vertices.

0.01 0.10 1 10 1000.325

0.33

0.335

0.34

0.345

0.35

0.355

0.36Line(3), Hawks in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.325

0.33

0.335

0.34

0.345

0.35

0.355

0.36Triangle, Hawks in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.24

0.25

0.26

0.27

0.28

0.29Complete(4), Hawks in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.24

0.25

0.26

0.27

0.28

0.29Square, Hawks in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.24

0.25

0.26

0.27

0.28

0.29Star(4), Hawks in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.24

0.25

0.26

0.27

0.28

0.29Diamond, Hawks in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.24

0.25

0.26

0.27

0.28

0.29Line(4), Hawks in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.24

0.25

0.26

0.27

0.28

0.29Paw, Hawks in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05



Dove fixation probability in a Hawk population

The fixation probabilities of a single Dove in a population of Hawks for smallgraphs on 3 and 4 vertices.

0.01 0.10 1 10 100

0.29

0.3

0.31

0.32

0.33

0.34Line(3), Doves in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 100

0.29

0.3

0.31

0.32

0.33

0.34Triangle, Doves in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.225

0.23

0.235

0.24

0.245

0.25

0.255

0.26

0.265Complete(4), Doves in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.225

0.23

0.235

0.24

0.245

0.25

0.255

0.26

0.265Square, Doves in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.225

0.23

0.235

0.24

0.245

0.25

0.255

0.26

0.265Star(4), Doves in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.225

0.23

0.235

0.24

0.245

0.25

0.255

0.26

0.265Diamond, Doves in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.225

0.23

0.235

0.24

0.245

0.25

0.255

0.26

0.265Line(4), Doves in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05

0.01 0.10 1 10 1000.225

0.23

0.235

0.24

0.245

0.25

0.255

0.26

0.265Paw, Doves in Hawk−Dove game

h

Ave

rage

fixa

tion

prob

abili

ty

v= 2 v= 1 v= 0.5 v= 0.25 v= 0.05


Discussion and future work

Outline

1 Credits










Discussion I

We have developed a new framework for modelling game theoreticalinteractions in a structured population.

The framework incorporates three key components: population structure;evolutionary dynamics; evolutionary game.

A useful area of application of our model is in animal territorialbehaviour.

Different species can exhibit different types of territoriality, and aflexible system of modelling this is required.



Discussion II

Evolutionary graph theory has made, and continues to make, importantcontributions to the understanding of the effect of population structure.

Our framework has some advantages over standard evolutionary graphtheory.

For example, multi-player games can be explicitly modelled in ourstructure.

In addition, there is a natural (and more logical) way of convertingaggregated game payoffs into fitness.



Discussion III

Following a recently accepted paper, we have considered an examplewhich incorporates all three key aspects of our framework.

In particular we have developed a birth-death dynamics and a naturalway to find the fixation probability of a rare mutant for any populationstructure.

We have seen that key features of the structure, including the temperatureand the mean group size, have a strong influence on the fixationprobability.

For example, high temperature amplifies the influences of the key gameparameters like the reward, increasing the fixation probabilities of thefitter individuals, and decreasing the fixation probabilities of the weakerindividuals.



Discussion IV

An important next step in this work is to more fully incorporateevolutionary dynamics in the new framework.

In evolutionary graph theory, there are a number of dynamics reflectingdifferent biological scenarios. In particular we need to developdeath-birth dynamics, as well as alternative birth-death dynamics.

Another aspect that needs development is the analysis of more realisticpopulations.

This requires larger populations, and also populations that have differingnumbers of places and individuals, i.e. which are not “graph-like”.

The above work is ongoing. We note that this research is still in itsrelatively early stages.


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