Figure 1.
Scenario A: The island population initially consists of individuals of genotype Ai Ai Bi Bi, whereas the mainland is homogeneous for genotype AmAmBmBm. Local selection acts on allele Ai on the island. After secondary contact is restored, genetic incompatibilities can occur between Ai and Bm. The model is extended by including Wolbachia in scenarios B–D. Colors indicate uninfected (white) and Wolbachia-infected populations (yellow and blue). Scenario B: Island infected with Wolbachia and mainland uninfected. Scenario C: Mainland infected with Wolbachia and no infection on the island before secondary contact. Scenario D: Before secondary contact, the island is infected with Wolbachia strain Wi and the mainland population with strain Wm. Scenarios B and C involve unidirectional CI, whereas scenario D involves bidirectional CI. After secondary contact, both nuclear and Wolbachia-induced cytoplasmic incompatibilities reduce number of offspring in intergroup matings.
Table 1.
Dobzhansky-Muller incompatibilities.
Table 2.
Wolbachia-induced cytoplasmic incompatibility (CI).
Figure 2.
Equilibrium frequencies of allele Ai on the island population as a function of the migration rate.
There are three qualitatively different outcomes. Circles show a typical example for the case without local selection (s = 0), boxes for the case with local selection (s>0) but without nuclear incompatibilities (lNI = 0), and triangles for the case in which both local selection (s>0) and nuclear incompatibilities (0<lNI≤1) act. The critical migration rate of nuclear divergence (mc,NI) is indicated by an arrow.
Figure 3.
Stability of Dobzhansky-Muller incompatibilities (DMI) for Scenario A (No Wolbachia).
Graph A: The critical migration rate (mc,NI) divides the parameter plane spanned by dominance level (h) and migration rate (m) into two regions. If m<mc,NI then alleles Ai and Bi can persist on the island, and genetic divergence at both loci is stable (green region). For all other parameter constellations, polymorphism at either locus is lost and DMI disappear (white region). The graph further shows that stability of genetic divergence increases with degree of genetic incompatibility. Parameters: lNI = 1 and s = 0.1. Graph B: Critical migration rates as functions of the selection coefficient for different dominance levels h = 0 (diamonds), h = 0.5 (boxes), h = 0.75 (black triangles), h = 0.99 (circles) and h = 1 (white triangles). Local selection has increasing effects on stability with increasing selection coefficient, and the largest changes occur for nearly dominant hybrid lethality (h = 0.99). Other parameter: lNI = 1.
Figure 4.
Stability of nuclear and cytoplasmic divergence for Scenario B (unidirectional CI with infected island population).
Graph A: Recessive Lethal NI. The parameter plane is divided into four zones: (1) stable NI and loss of Wolbachia (orange), (2) stable NI and Wolbachia persistence (yellow), (3) loss of NI and Wolbachia persistence (blue), (4) loss of both NI and Wolbachia (white). The dotted and solid lines indicate the critical migration rates of NI and CI, respectively. Note that Wolbachia causes a fecundity reduction of f = 0.1. Other parameters: lNI = 1, h = 0, s = 0.1. Graph B: Variable Dominance NI. Critical migration rates of nuclear divergence (mc,NI) as function of the CI level and for different dominance levels h = 0 (black triangles), h = 0.5 (boxes), h = 0.99 (circles) and h = 1 (empty triangles). Parameters: lNI = 1, s = 0.1, f = 0.1.
Figure 5.
Stability of nuclear and cytoplasmic divergence for Scenario C (unidirectional CI with infected mainland population).
Graph A: Recessive Lethal NI. Critical migration rates of NI (dotted line) and CI (solid line) as functions of the CI level. As in figure 4, the parameter space is divided into four zones. CI collapse is either caused by the spread of Wolbachia on the island (lCI>0.1) or by Wolbachia loss in both populations (lCI<0.1). Other parameters: lNI = 1, h = 0, s = 0.01, f = 0.1. The graph shows that unidirectional CI has only weak effect on the stability of recessive NI. Graph B: Variable Dominance NI. Critical migration rates for nuclear divergence as functions of the CI level for different dominance levels h = 0 (diamonds), h = 0.5 (boxes), h = 0.75 (daggers), h = 0.99 (circles) and h = 1 (triangles). Parameters: lNI = 1, s = 0.01, f = 0.1. Unidirectional CI has the largest effects on stability of NI when the CI level is high and dominance is near perfect, but has some leakage (e.g. h = 0.99). In this case, CI and NI act synergistically in the sense that both CI and NI reach much higher critical migration rates than either alone in the corresponding mainland-island models.
Figure 6.
Stability of nuclear and cytoplasmic divergence for Scenario D (bidirectional CI).
Graph A: Recessive Lethal NI. Bidirectional CI increases stability of nuclear divergence. The graph shows that the parameter plane is divided into four zones: (1) stable NI and loss bidirectional CI (orange), (2) stable NI and bidirectional CI (yellow), (3) loss of NI and stable bidirectional CI (blue), (4) loss of both NI and bidirectional CI (white). The dotted and solid lines indicate the critical migration rates of NI and bidirectional CI, respectively. Bidirectional CI is stable if both Wolbachia strains stably coexist. Parameters: lNI = 1, h = 0, s = 0.1. Under these parameter values of strong recessive NI, presence of NI has little effect on CI stability, whereas CI level strongly effects stability of nuclear divergence. Graph B: Variable Dominance NI. Critical migration rates for nuclear divergence as functions of the CI level for h = 1 (triangles), h = 0.99 (circles), h = 0.5 (boxes) h = 0 (diamonds). Other parameters: lNI = 1, s = 0.1. Whereas CI has no effect when lethality dominance is 1 (as expected), it has a large effect when even a low level of F1 hybrid survival occurs (h = 0.99).
Figure 7.
Analytical approximations of critical migration rates.
Squares indicate Scenarios B (unidirectional CI with infected island population), and circles Scenario D (bidirectional CI). Black and white symbols represent numerical results and analytical approximations using formula (1), respectively. Graph A: Recessive Lethal NI. Parameters: lNI = 1, h = 0, s = 0.1. Graph B: Dominant Lethal NI. Parameters: h = 0.99, lNI = 1, s = 0.1. Approximations are especially good for unidirectional CI and low to moderate bidirectional CI.