Figure 1.
Invasion profiles of the two morphs.
These show the disperser morph (dashed line) and establisher morph (solid line) when present in the landscape on their own (rows (i) and (ii)) and when mutation allows both to be present (row (iii)). The simulations were initiated with the first 100 cells occupied by each phenotype at its equilibrium population density and the remaining cells unoccupied. The simulations were run on a lattice consisting of 15000 cells. For all graphs each line represents the density profiles at a different time point, with each time point 500 units apart. In column (a) the polymorphic invasion speed is the same as the monomorphic establisher speed; in column (b) the polymorphic invasion speed is the same as the monomorphic disperser speed, and in (c) the polymorphic invasion speed is faster than either monomorphic invasion. For all simulations ,
, and in (a)
,
,
,
; (b)
,
,
,
; (c)
,
,
,
.
Figure 2.
Comparison of analytical and numerical predictions of the invasion speed.
This is an example of the case when polymorphism results in faster invasions than either single morph. The circles represent numerical results calculated from simulations, and the curve numerical predictions found using the van Saarloos method [27]. Parameter values used were ,
,
,
,
and the values of
used for the numerical simulations were 0.5, 0.6, 0.7, 0.8 and 0.9.
Figure 3.
Comparison of stochastic and deterministic invasion speeds at different carrying capacities.
The filled symbols represent the deterministic prediction and the empty symbols the stochastic predictions. The triangles represents the establisher morphs speed, the crosses the disperser morphs speed and the circles the invasion speed when both morphs are present. In (a) the polymorphic invasion speed is the same as the monomorphic establisher speed; in (b) the polymorphic invasion speed is the same as the monomorphic disperser speed, and in (c) the polymorphic invasion speed is faster than either monomorphic invasion. Parameter values used in (a) and (b) are the same as in Fig. 1 (a) and (b) and in (c) the parameter values used were ,
,
,
,
, with
ranging from
to
.
Figure 4.
Comparison of invasion speeds with different values of .
The parameter values are the same as in Fig 3(c) with the value of varied from
(crosses),
(diamonds),
(circles),
(plus) and
(triangles point down). The filled circle represents the polymorphic deterministic speed which is the same for all mutation rates. The triangles point up represent the fastest single morphs speed, which here is the establisher, with the filled triangle the deterministic speed.
Figure 5.
Anomalous invasion speeds when individual morph speeds are similar.
The triangles represents the establisher morphs speed, the crosses the disperser morphs speed and the circles the invasion speed when both morphs are present. Parameter values used were ,
,
,
,
and
ranges from
to
.