Figure (2.6) compares the theoretical equilibria derived from the mean-eld (equation (2.3)) and pair (equation (2.14)) ODE analysis with the stochastic simulation data for these two models at m = 4. With a population of only 2500 individuals there is clearly good agreement between the simulations and ODE systems, and with this hindsight the continuity assumptions can be justied.
0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s=v/c fraction doves
Figure 2.6: Comparison of pair (red) and mean-eld (blue) simulation data (irreg- ular lines - from gure (2.5)) with the corresponding theoretical equilibria (straight lines). Also shown for the pair model is the fraction of hawk-dove pairs in the total number of pairs (black) for the simulation and theory and the non-spatial theoretical equilibrium (dotted).
Not surprisingly, the largest discrepancy between the ODE and simulations occurs when both predict a small proportion of hawks or a small proportion of doves. The tendency is for a small population to become extinct much more readily in the stochastic models, as was observed in the simulations, whereas deterministic ODE equilibria do not suer this fate. For predicting the general trend of the simulations this causes no problem, but it cannot be ignored in the prediction of a particular outcome when a small population is expected. For example, in the pair model at s = 0:3 the ODE predicts an equilibrium population of 10% hawks but the stochastic simulation, run over an interme- diate time-scale may well eliminate hawks all together. Of course in a larger population (larger N), one expects, and nds, this disagreement is reduced. Small fractions are then represented by larger numbers of individuals and the expected time to extinction is much greater. (However it is worth noting that in any such nite stochastic system an absorbing state of extinction for one species will ultimately always be reached, though possibly over irrelevantly large times-scales). Whilst keeping
in mind these concerns of demographic stochasticity, we should not lose sight of the fact that the qualitative agreement is clearly close enough to merit continued study of the ODE systems. It is interesting to note the dependency of the mean-eld and pair ODE equilibria on m, the neigh- bourhood size. The equations for both the straight lines in gure (2.6) imply that both `swing round' to approach the line r = 1;s (and each other) as m! 1 the mean-eld line pivots at
s = 0r = 1 and the pair line pivots in the middle at s = 0:5r = 0:5. As individuals interact with larger and larger neighbourhoods, they see a larger and statistically less variable proportion of the total population. In the limit, everyone interacts with everyone and we arrive at the non-spatial case. Both mean-eld and pair models therefore capture one interesting aspect of the nature of the spatial game - the importance of the small size of the interaction neighbourhood. The mean-eld model in particular shows how important this can be in comparison to the usual assumption of complete mixing of the population (c.f. the original hawk-dove game). It is also clear that the pair model is able to incorporate other spatial eects, connected with correlations between neighbours, that the mean-eld does not and these too have signicant eect on the model behaviour.
In this example, the pair model appears to faithfully reproduce most of the essential behaviour of the interactions seen in the lattice model - at least as far as discernible using the pair variables - although the importance of any further secondary structure to the dynamics is unclear. The question of whether the lattice model itself is a useful or naive model remains unanswered it is certainly very restrictive in its treatment of spatial relations and alternatives will be discussed in later chapters. It may be argued that pair models alone provide a better framework for a spatial model in some biological systems. But for the moment their use is conned to tools for understanding more explicit systems like the lattice model as models of models.
2.6 General
2 2Games
It is possible to extend the Pair model analysis of section (2.4.2) to consider a general payo matrix for an identical two player, two strategy game. Label the strategies x and y and write the payo matrix as species x species y 0 B B B @ x y a b c d 1 C C C A (2.15) Again, the only restrictions on the entries are that all must be non-negative. Proceeding as before, the expected tnesses Fx and Fy, respectively of an x individual neighbouring at least one y and a
y neighbouring at least one x are now Fx = a(m;1) m PxyP+ Pxx xx + b 1m + (m;1) m (PxyP+ Pxy xx) = m(Pxy1+ Pxx) (a(m;1)Pxx+ b(mPxy+ Pxx)) Fy = c 1m + (m;1) m (PxyP+ Pxy yy) + d(m;1) m PxyP+ Pyy yy = m(Pxy1+ Pyy) (d(m;1)Pyy+ c(mPxy+ Pyy)) (2.16)
The rates are identical to equations (2.6). The algebra involved in solving for the equilibrium in this case is predictably longer and the nal forms given below were derived using MAPLE (Appendix B). For convenience, N was scaled to one. In addition to the segregated equilibria at Pxy=0Pxx+Pyy=
m, the following non-trivial xed points were found.
solution1 Pxx = m 2(d ;b)((d;b)(m;1);(a;c)) (m;1)(m;2)(a;c + d;b) 2 Pyy = m 2(a ;c)((a;c)(m;1);(d;b)) (m;1)(m;2)(a;c + d;b) 2 Pxy = m((a;c)(m;1);(d;b))((d;b)(m;1);(a;c)) (m;1)(m;2)(a;c + d;b) 2 (2.17) x = (d;b)(m;1);(a;c) (m;2)(a;c + d;b) y = (a;c)(m;1);(d;b) (m;2)(a;c + d;b) solution2 Pxx = bm 2(d(m ;1) + c)
(m;1)(m(ac + bd;2ad);2(a;b)(c;d))
Pyy = cm
2(a(m
;1) + b)
(m;1)(m(ac + bd;2ad);2(a;b)(c;d))
Pxy = ;m(a(m;1) + b)(d(m;1) + c)
(m;1)(m(ac + bd;2ad);2(a;b)(c;d))
(2.18) x = (b;a)(d(m;1) + c)
m(ac + bd;2ad);2(a;b)(c;d)
y = (c;d)(a(m;1) + b)
m(ac + bd;2ad);2(a;b)(c;d)
Solution 2 can be eectively discounted as it is unphysical (0PxxPxyPyym and 0xy1
do not hold) in almost all cases: Lemma
There exists only one non-trivial physically meaningful solution of form (2.18) for payo matrices of type (2.15), which is given by Pxx= Pyy= 0Pxy= m=2 in the case b = c = 0.
Proof
Suppose there is such a solution with abcd0 and m > 1. We have
(m;1)d + c 0
(m;1)a + b 0
Let D = (m;1)(m(ac + bd;2ad);2(a;b)(c;d)) be the denominator of PxxPxyand
Pyy in equation (2.18).
If Pxx> 0, (2.18))D > 0)Pxy< 0, a contradiction so Pxx= 0.
If Pyy> 0, (2.18))D > 0)Pxy< 0, a contradiction so Pyy= 0.
This leaves the unique solution Pxx= Pyy= 0Pxy= m=2 obtained when b = c = 0. 2
Furthermore, solution 2 is unstable: Calculating the eigenvalues of the Jacobian matrix of partial derivatives using MAPLE found them to be (m;1)a and (m;1)d, both of which are positive for
non-trivial payo matrices.
Solution 1 is more interesting. It depends only on the values a;c and d;b, not abc and d
individually, so substitute = a;c and = d;b. We can then nd restrictions on the payo
parameters and which admit meaningful solutions with Pxy6= 0. Note it is sucient to further
check Pxx and Pyy are non-negative, because then 0PxxPxyPyy m and 0 < SxSy < 1 are
guaranteed. For its existence, we must assume m > 2 and + 6= 0 in the remainder of this section.
The rst assumption is a natural one to make, for the only population with just two neighbours per individual is necessarily strung out along a rather unnatural one-dimensional line. The second assumption eliminates a special but uninteresting case when one species outcompetes the other. Now suppose > 0 and we have such a solution. From (2.17),
Pyy0,(m;1) but (m;1) and Pxy> 0 , 1 (m;1) so we must have 1 (m;1) (m;1) (2.19)
and Pxx0 follows. Similarly, if < 0 we nd that
(m;1)
1 (m;1)
(2.20) Figure (2.7) shows these restricted regions on the plane.
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 a-c d-b P1 P2
Figure 2.7: A plot of the plane showing the region (shaded) in which solution 1 represents a non-trivial and meaningful equilibrium solution. The dashed line is the singular case + = 0. The dotted line is that corresponding to the original hawk-dove game points P1 = (
1
mm;1
m ) and P2 = (
m;1
m 1
m) are the limits of the
solution (cf. gure (2.5)).
2.6.1 Stability
Calculating the stability of the xed points is slightly more complicated in the general case, but lin- earisation about the equilibrium using the Jacobian matrix yields the following result about solution 1:
Lemma
When it exists as a meaningful solution, in the sense of gure (2.7), solution 1 is stable for < 0 and unstable for > 0.
Proof
Using MAPLE the two eigenvalues are found to be U p V where U = ; 1 2(m;2) (m;1) (m + 2m( b + c);( + )(b + c)) ( + ) V = U2+ 2(m ;2) (m;1) ( + b + c)( (m;1);)((m;1); ) ( + )2
The equation for Pxy implies ( (m;1);)((m;1); ) is always positive, because Pxy
must be positive. Therefore V U
2 if and only if
+ b + c = b + a0
There are two cases:
i. > 0 )V > U
2and both eigenvalues are real, but are of opposite sign. The equilib-
rium is therefore an unstable saddle-point.
ii. < 0 ) V < U
2 so either both eigenvalues are real and of the same sign as U, or
they are a complex conjugate pair with real part U. Both cases are stable if and only if U < 0, and we show that this must be the case:
The sign of U is the sign of
! = m + 2m( b + c);( + )(b + c)
= b((2m;1) ;) + c((2m;1); ) + m
Equation (2.20) implies (2m;1) ; < 0 and recall abcd0) ;c )m ;mc, so
! < 0 + c((2m;1); );mc
= c((m;1); )
0 using (2.20) again 2
By similar analysis, it is possible to show that when the coexistence equilibrium is unstable, both the single species solutions (all x or all y) are themselves locally stable, and the eventual equilibrium position depends on the initial conditions.