Effects of Host Interspecific Interaction in the Maculinea–Myrmica Parasite–Host System

Abstract

A model of interspecific host competition in a system with one parasite (butterfly—Maculinea) and multiple potential hosts (ants—Myrmica) is presented. Results indicate that host interspecific competition increases the occurrence of multiple host behaviour in Maculinea natural populations but decreases the ability of the parasite populations to adapt to the most abundant host species. These qualitative predictions were compared with data on host specificity, with good agreement. Analysis of the data also indicates that Maculinea teleius and Maculinea arion respond differently to changes in relative host abundances. Maculinea teleius shows a larger fraction of sites where it displays multiple host behaviour and a larger fraction of sites where the niches of the hosts overlap. In some instances, Maculinea teleius is adapted to Myrmica hosts that are present in lower frequencies. Maculinea arion is locally more host-specific and occurs at sites where host interspecific competition is unlikely and is more frequently adapted to the most abundant host species.

Appendices

Appendix A: Testing the Significance of the Patterns

In Sect. 3.2 we defined two classes of interspecific competition, C and NC, which are based on the relations between ecological niches of 11 species. Each pair of species was classified to be either in a state of “higher chance of competition” or “smaller chance of competition”, using the information contained in the literature (Elmes et al. 1998; Radchenko and Elmes 2010) about the niche of each species. If there was an estimate of a significant niche overlap [see Fig. 1 in Elmes et al. (1998)], the species were said to be in class C, while if niche overlap was estimated as small or non-existent, the species we assigned to the disjoint class NC.

Fig. 7

For each simulation, a random matrix $\hbox {RM}_\mathrm{C}$ is generated and the total score of V’s (as in Table 3) is computed. The figure represents the frequencies of the obtained scores if the relations between the species were randomly attributed. Of the total 1000 simulations none achieved the score of 30/30, attained by the matrix $M_\mathrm{C}$, indicating the low probability that the patterns observed are due to chance

The matrix $M_\mathrm{C}$ (see Table 4) is used to separate the data from the sites into categories, which are then analysed for the patterns (observations 1–3, in Sect. 3) described by the theoretical model. Therefore, to test whether the patterns did not arise simply by chance, we simulated the results that would be obtained if the symmetric binary matrix $M_\mathrm{C}$ was created using random associations between the Myrmica species (that is, that do not take into account the biological information on niches).

$M_\mathrm{C}$ has 14 nonzero elements and the diagonal is zero. Since it is symmetric, to create a correspondent random matrix, it is enough to generate 7 entries equal to 1 above the diagonal. If these entries are placed at random, the matrix thus obtained represents a random association of Myrmica species, not based on ecological properties. The random matrix represents an arbitrary classification of the Myrmica species.

We simulated 1000 random associations, and for each, we repeated the analysis presented in Table 3, calculating the total number of agreements between theoretical predictions and empirical data. For these cases where some statistic or test could not be calculated (if the class had no members or not enough to calculate a correlation, for example), the test was counted as positive, that is the evaluation of the scores of the random matrices was biased in favour of the random attributions. In none of the 1000 simulations we obtained a score equal to the one obtained by the matrix $M_\mathrm{C}$. In Fig. 7 we present a histogram with the distribution of results.

The results from these simulations strongly indicate that the agreement with all three theoretical observations has a very small probability of being obtained by chance, meaning that the patterns are robust.

Appendix B: Mathematical Analysis

Change of variables in Eq. (2): $y_1=p_1/(p_1+p_2)$, $y_2=p_2/(p_1+p_2)=1-y_1$. Results in a three-dimensional system:

$$\begin{aligned} \frac{{\hbox {d}}h_1}{{\hbox {d}}t}= & {} \lambda _1h_1\left( 1-h_1- c h_2 -\gamma _1 y_1\right) \nonumber \\ \frac{{\hbox {d}}h_2}{{\hbox {d}}t}= & {} \lambda _2h_2\left( 1-h_2- c h_1 -\gamma _2 (1-y_1)\right) \nonumber \\ \frac{{\hbox {d}}y_1}{{\hbox {d}}t}= & {} y_1(1-y_1) \frac{\gamma _1 h_1-\gamma _2 \sigma h_2}{\gamma _1h_1+\gamma _2 \sigma h_2} . \end{aligned}$$

(3)

We will exclude equilibria with both $h_1=h_2=0$, but numerical simulations with a model that includes small perturbation in the denominator ($\gamma _1h_1+\gamma _2 \sigma h_2+\epsilon$) present results very similar to the original, indicating that the singularity is not a problem for modelling purposes. Also, if both populations were absent, the ecological problem loses its meaning, since in the absence of hosts the parasite cannot be present.

There are 6 equilibria for $h_1+h_2>0$:

$A=(0,h_{2A},0)$, with $h_{2A}=1-\gamma _2$, feasible for

$$\begin{aligned} \gamma _2 \le 1. \end{aligned}$$

(4)

$B=(0,h_{2B},y_{1B})$, with $h_{2B}=1$, $y_{1B}=1$, always feasible.

$F=(h_{1F},0,0)$, with $h_{1F}=1$, always feasible.

$C=(h_{1C},0,y_{1C})$, with $h_{1C}=1-\gamma _1$, $y_{1C}=1$, feasible for

$$\begin{aligned} \gamma _1 \le 1. \end{aligned}$$

(5)

$D=(h_{1D},h_{2D},0)$, with

$$\begin{aligned} h_{1D}=\frac{1-c+c\gamma _2}{1-c^2}, \quad h_{2D}=\frac{1-c-\gamma _2}{1-c^2}, \end{aligned}$$

feasible for one of the alternative sets of conditions

$$\begin{aligned}&c^2 < 1, \quad 1 \ge c+\gamma _2, \quad 1+c\gamma _2 \ge c; \end{aligned}$$

(6)

$$\begin{aligned}&c^2 > 1, \quad 1 \le c+\gamma _2, \quad 1+c\gamma _2 \le c. \end{aligned}$$

(7)

Finally we have the coexistence point

$E=(h_{1E},h_{2E},y_{1E})$ that gives rise to two different cases.

$E_a=(h_{1Ea},h_{2Ea},y_{1Ea})$ with $y_{1Ea}=1$,

$$\begin{aligned} h_{1Ea}=\frac{1-c-\gamma _1}{1-c^2}, \quad h_{2Ea}=\frac{1-c+c\gamma _1}{1-c^2}. \end{aligned}$$

Alternatively, we can impose $\gamma _1 h_1 = \gamma _2 \sigma h_2$ to satisfy the third equilibrium equation, and in such case we obtain

$$\begin{aligned} h_{1Eb}= & {} \frac{\gamma _1 \gamma _2 \sigma - \gamma _1 \gamma _2^2 \sigma + \gamma _2^2 \sigma }{\gamma _1^2 + c \gamma _1 \gamma _2 + \gamma _2^2 \sigma + c \gamma _1 \gamma _2 \sigma },\quad h_{2Eb}=\frac{\gamma _1^2 - \gamma _1^2 \gamma _2 + \gamma _1 \gamma _2}{\gamma _1^2 + c \gamma _1 \gamma _2 + \gamma _2^2 \sigma + c \gamma _1 \gamma _2 \sigma },\\ y_{1Eb}= & {} \frac{1}{\gamma _1} (1-h_{1Eb}-ch_{2Eb}) =\frac{\gamma _1 - \gamma _2 \sigma - c \gamma _1 + c \gamma _2 \sigma + c \gamma _1 \gamma _2 + \gamma _2^2 \sigma }{\gamma _1^2 + c \gamma _1 \gamma _2 + \gamma _2^2 \sigma + c \gamma _1 \gamma _2 \sigma }. \end{aligned}$$

The Jacobian of (3) is

$$\begin{aligned} J= \left( \begin{array}{ccc} J_{11} &{} -c \lambda _1 h_1 &{} - \gamma _1 \lambda _1 h_1\\ -c \lambda _2 h_2 &{} J_{22} &{} \gamma _2 \lambda _2 h_2\\ J_{31} &{} J_{32} &{} J_{33} \end{array} \right) \end{aligned}$$

with

$$\begin{aligned} J_{11}= & {} \lambda _1 (1-2h_1 -ch_2 -\gamma _1 y_1), \quad J_{22}=\lambda _2 [1-2h_2 -ch_1 -\gamma _2 (1-y_1)],\\ J_{31}= & {} 2 y_1 (1-y_1) \frac{\sigma \gamma _1 \gamma _2 h_2 }{(\gamma _1 h_1+ \gamma _2 \sigma h_2)^2},\quad J_{32}= -2 y_1 (1-y_1) \frac{\gamma _1 \gamma _2 h_1 \sigma }{(\gamma _1 h_1+ \gamma _2 \sigma h_2)^2},\\ J_{33}= & {} (1-2y_1) \frac{\gamma _1 h_1 - \gamma _2 \sigma h_2 }{\gamma _1 h_1 + \gamma _2 \sigma h_2}. \end{aligned}$$

At A the eigenvalues of the Jacobian evaluated at this point J(A) are $-1$, $\lambda _1 (1-c h_{2A})$, $-\lambda _2 h_{2A}$, entailing the stability conditions $1<c h_{2A}$ and $-\lambda _2 h_{2A}<0$ or, extensively,

$$\begin{aligned} 1+c\gamma _2<c , \quad \gamma _2 < 1. \end{aligned}$$

(8)

For the equilibrium B we find $-\lambda _2$, $-\lambda _1 (\gamma _1 +c-1)$ and 1, so its unconditional stability is obtained.

For the equilibrium F we find instead $-\lambda _1$, $\lambda _2 (1-c - \gamma _2)$ and 1, so the equilibrium is also unconditionally stable.

The point C again leads to an immediate evaluation of the eigenvalues: $-\lambda _1 h_{1C}$, $\lambda _2- c \lambda _2 h_{1C}$, $-1$, giving a stable equilibrium for

$$\begin{aligned} h_1> \max \left\{ \frac{1-\gamma _1}{2}, \frac{1}{c} \right\} . \end{aligned}$$

which explicitly become

$$\begin{aligned} \gamma _1 < 1 - \frac{1}{c}. \end{aligned}$$

(9)

The Jacobian at the equilibrium D has the explicit eigenvalue

$$\begin{aligned} \frac{\gamma _1 h_{1D} - \gamma _2 \sigma h_{2D} }{\gamma _1 h_{1D} + \gamma _2 \sigma h_{2D}} \end{aligned}$$

from which the first stability condition

$$\begin{aligned} \gamma _1 h_{1D} < \gamma _2 \sigma h_{2D}. \end{aligned}$$

(10)

The Routh–Hurwitz conditions on the remaining 2 by 2 minor are

$$\begin{aligned} \lambda _1 h_{1D} + \lambda _2 h_{2D}>0,\quad (1-c^2) h_{1D} h_{2D} >0, \end{aligned}$$

which explicitly can be rewritten as

$$\begin{aligned} \lambda _1 + c \lambda _1 \gamma _2 + \lambda _2 < c\lambda _1 + c \lambda _2 + \lambda _2 \gamma _2, \quad c^2 > 1 \end{aligned}$$

or

$$\begin{aligned} \lambda _1 + c \lambda _1 \gamma _2 + \lambda _2 > c\lambda _1 + c \lambda _2 + \lambda _2 \gamma _2, \quad c^2 < 1 \end{aligned}$$

and

$$\begin{aligned} c {\gamma _2}^2 + c^2 \gamma _2 + \gamma _2 + 2 c> 2c \gamma _2 + c^2 + 1, \quad c^2 >1 \end{aligned}$$

or

$$\begin{aligned} c {\gamma _2}^2 + c^2 \gamma _2 + \gamma _2 + 2 c< 2c \gamma _2 + c^2 + 1, \quad c^2 <1 \end{aligned}$$

At coexistence $E_a$, the Jacobian simplifies a bit; we find

$$\begin{aligned} J(E_a)= \left( \begin{array}{ccc} -\lambda _1 h_{1Ea} &{} -c \lambda _1 h_{1Ea} &{} - \gamma _1 \lambda _1 h_{1Ea}\\ -c \lambda _2 h_{2Ea} &{} -\lambda _2 h_{2Ea} &{} \gamma _2 \lambda _2 h_{2Ea}\\ 0 &{} 0 &{} -1 \end{array} \right) \end{aligned}$$

while at $E_b$ instead we have

$$\begin{aligned} J(E_b)= \left( \begin{array}{ccc} -\lambda _1 h_{1Eb} &{} -c \lambda _1 h_{1Eb} &{} - \gamma _1 \lambda _1 h_{1Eb}\\ -c \lambda _2 h_{2Eb} &{} -\lambda _2 h_{2Eb} &{} \gamma _2 \lambda _2 h_{2Eb}\\ y_{1Eb} (1-y_{1Eb}) \frac{\gamma _1 }{2 \gamma _2 \sigma h_{2Eb}} &{} -y_{1Eb} (1-y_{1Eb}) \frac{1}{2 h_{2Eb}} &{} 0 \end{array} \right) \end{aligned}$$

For $E_a$ we have one eigenvalue, and by Routh–Hurwitz conditions on the remaining minor, the one on the trace is satisfied

$$\begin{aligned} -\,{\text {tr}}(J(E_a)) = \lambda _1 h_{1Ea} + \lambda _2 h_{2Ea} >0 \end{aligned}$$

(11)

which explicitly can be rewritten as

$$\begin{aligned} \lambda _1 + \lambda _2 + c \lambda _2 \gamma _1 > c \lambda _1 + \lambda _1 \gamma _1 + c \lambda _2, \quad c^2 <1 \end{aligned}$$

or

$$\begin{aligned} \lambda _1 + \lambda _2 + c \lambda _2 \gamma _1 < c \lambda _1 + \lambda _1 \gamma _1 + c \lambda _2, \quad c^2 >1, \end{aligned}$$

while for the determinant we find

$$\begin{aligned} \det (J(E_a)) = (1-c^2) \lambda _1 h_{1Ea} \lambda _2 h_{2Ea}>0 \end{aligned}$$

which is positive only if we require

$$\begin{aligned} 2c+c^2 \gamma _1+\gamma _1+c \gamma _1^2> 1+c^2+2c \gamma _1, \quad c^2>1 \end{aligned}$$

or

$$\begin{aligned} 2c+c^2 \gamma _1+\gamma _1+c \gamma _1^2< 1+c^2+2c \gamma _1, \quad c^2<1. \end{aligned}$$

At $E_b$ the Jacobian is almost full, but the condition on the trace is the same as above (11). For the determinant we find

$$\begin{aligned} -\det (J(E_b))= \frac{y_{1Eb} (1-y_{1Eb})}{2 \gamma _2 \sigma } \lambda _1 h_{1Eb} \lambda _2 [\gamma _1^2 + c \gamma _1\gamma _2 +c\gamma _1 \gamma _2 \sigma +\gamma _2^2 \sigma ] \end{aligned}$$

which is positive only if we require

$$\begin{aligned} y_{1Eb}<1. \end{aligned}$$

(12)

Further, the condition on the sum of minors leads to the following inequality

$$\begin{aligned}&(\lambda _1 h_{1Eb} + \lambda _2 h_{2Eb})\left[ \lambda _1 h_{1Eb} \lambda _2 h_{2Eb} (1 -c^2)\right. \nonumber \\&\qquad \left. +\,y_{1Eb}(1- y_{1Eb}) \left( \frac{\gamma _1 (\gamma _1 \lambda _1 h_{1Eb})}{2 \gamma _2 \sigma h_{2Eb}} + \frac{\gamma _2 \lambda _2 h_{2Eb}}{2 h_{2Eb}} \right) \right] \nonumber \\&\quad > \frac{y_{1Eb} (1-y_{1Eb})}{2 \gamma _2 \sigma } \lambda _1 h_{1Eb} \lambda _2 [\gamma _1^2 + c \gamma _1\gamma _2 +c\gamma _1 \gamma _2 \sigma +\gamma _2^2 \sigma ]. \end{aligned}$$

(13)

For a Hopf bifurcation, condition (13) becomes an equality, for some chosen bifurcation parameter.

Appendix C: Parameter Estimation

The parameters to be estimated are: selection pressures ($\gamma _i$), relative rates of reproduction ($\lambda _i$), the relative abundance of hosts ($\sigma$) and maximum impact of interspecific competition (c).

The selection pressures, $\gamma _i$, represent the maximum impact on fitness by the population of parasites on one host. For the Maculinea–Myrmica system it is limited by several factors, such as the percentage of area that the food plants occupy in the habitat of the host. This limited distribution of the food plants limits the fraction of hosts exposed to the parasite, given that the maximum foraging distance of ant colonies is around 2 m (Elmes et al. 1998). There is also a maximum parasite impact on the reproduction rate of the host, because most nests survive parasite exploitation and, in some cases, part of the ant brood is not available to be parasitized (Thomas and Wardlaw 1992). In Griebeler and Seitz (2002) the average of the area covered by the food plants (Thymus pulegioides) at different sites was less than 11.75% (minimum of 1%, maximum of 10%, SD 11.56%). Hochberg et al. (1994) used values between 50 and 70% for the percentage of nests exposed to the parasite, and Elmes and Wardlaw (1982) have shown that, at a site where Maculinea arion was present, colonies of Myrmica sabuleti had an average of 183 workers, while at a site where the parasite was absent the average was of 317 workers. For the same site, Thomas and Wardlaw (1992) found an average nest size of 180 (SD 164) workers. Given that the number of workers is closely correlated with the production of reproductive female individuals (Elmes 1973; Elmes and Wardlaw 1982), we can estimate that the average maximum impact on fitness for that particular site is below 43%. All the above information suggests that parasite pressure, in general, is below 0.7.

The relative reproduction rates $\lambda _i=r_i/r_\mathrm{P}$ can be approximated using the estimates of $R_0$, which in Elmes et al. (1996) denotes the maximum yearly reproduction rate of the parasite species, for the parasite and host species (in logistic growth, $r_\mathrm{P}\approx R_0$). In a modelling study, Hochberg et al. (1992) estimate $R_0$ as 6.2 for Maculinea rebeli, while Thomas et al. (2009) have an estimate of 5.49 for Maculinea arion. For the Myrmica species, Elmes (1973) uses a value of 1.6 for Myrmica rubra, while Hochberg et al. (1994) in a model with various Myrmica species adopts an interval between 1.3 and 2 (depending on environmental factors) for $R_0$. These estimates suggest a value of $\lambda _i$ in the interval 0.21–0.36 (a more conservative interval is from 0.14 to 0.40, using $r_{\mathrm{logistic}}\approx \ln (R_0)$).

The relative abundance of hosts ($\sigma =K_2/K_1$) is the carrying capacity of the site in relation to host 2 divided by the carrying capacity in relation to host 1. The studies quoted in Sect. 2.2 are usually conducted within the range of the food plants, and studies where the distribution of Myrmica nests outside this range are rare. Therefore, one can only work with hypotheses on the selection pressures and use the estimates of $K_i\gamma _i$, which is actually what is measured by looking for host nests within the area close to food plants, to estimate $\sigma$. For instance, if selection pressures are approximately equal, it is enough to divide the frequency of nests of host 2 found inside the food plant area by the frequency of nests of host 1 ($M_2/M_1$) to estimate $\sigma$.

The maximum effect of interspecific competition (c) is the maximum fraction of population size that may be reduced due to the presence of other host species. It depends on several factors: habitat distribution, behavioural variations among species and climatic variation (Hölldobler and Wilson 1990; Elmes et al. 1998; Radchenko and Elmes 2010). Hochberg et al. (1994) have worked with a habitat superposition of 50%, but the great variability of the relative abundance of different species (as illustrated in Sect. 2.2) suggests that the overlapping patterns may be more variable and unpredictable than the simple gradient pattern used in the models. In our simulations, we allow c to vary in the interval (0, 1), both to incorporate this variability and as a way to better explore the effects of interspecific competition.

Appendix D: Random Parameters and Initial Conditions

Given the range of parameters presented in Table 2, to investigate the influence of interspecific competition on host specificity, we chose 11 values of c in the set $S=\{0,0.1,\dots ,1\}$ and, for each value of c, took random combinations of parameters and initial conditions according to the rules given below:

• $\gamma$: uniform distribution within its range: [0.01, 0.7].

• $\lambda$: uniform distribution within its range: [0.14, 0.4].

• $p_1(0)$, $p_2(0)$, $h_1(0)$ and $h_2(0)$: uniform in [0, 1].

• $T_\mathrm{H}=K_1/(K_1+K_2)=1/(1+\sigma )$: uniform in [0, 1].

The reason for sampling $T_\mathrm{H}$ instead of $\sigma$ is to generate the various scenarios of different host abundance, without bias.

For each different value of c, 1000 trials were run, each with a new set of random parameters and initial conditions. Of the total 11000 simulations, 790 ($\approx 7\%$) failed numerically and were discarded. In Figs. 8 and 9 we present results of 100 simulations for each value of c in $\{0,0.2,0.4,0.6,0.8,1\}$.

Fig. 8

Relation between the theoretical host abundance $T_\mathrm{H}=K_1/(K_1+K_2)=1/(1+\sigma )$ and the theoretical parasite distribution $T_\mathrm{P}=P_1/(P_1+P_2)$. Simulations of model 2 for random $\gamma$, $\lambda$ and initial conditions $h_1(0),h_2(0),p_1(0)$ and $p_2(0)$. 100 points for each value of $c \in \{0, 0.2, 0.4\}$

Fig. 9

Relation between the theoretical host abundance $T_\mathrm{H}=K_1/(K_1+K_2)=1/(1+\sigma )$ and the theoretical parasite distribution $T_\mathrm{P}=P_1/(P_1+P_2)$. Simulations of model 2 for random $\gamma$, $\lambda$ and initial conditions $h_1(0),h_2(0),p_1(0)$ and $p_2(0)$. 100 points for each value of $c \in \{0.6, 0.8, 1\}$

Fig. 10

Plot of c versus the slopes of regressions between $T_\mathrm{P}$ and $T_\mathrm{H}$ (each calculated for a different value of c). The coefficient decreases as the effect of interspecific competition increases

Figures 8 and 9 indicate that the observations made for the simulations performed with fixed parameters are also valid for the simulations with random parameters, that is, even with conditions changing from site to site, the patterns observed should still hold. To test more precisely observation 1 we calculated the regression slope for $T_\mathrm{H}$ and $T_\mathrm{P}$ in the simulations for each value of $c \in S$. Results are presented in Fig. 10.

To support observation 2, we calculated the ratio of points in the region $(0,0.5)\times [0.5,1]\cup (0.5,1)\times [0,0.5]$ over the total points for each value of c. It represents the simulations where the primary host was less abundant than the secondary. The simulations with $c=1$ showed approximately 50% of such occurrence, with $c=0.9$ around 3%, and for the other values of c, percentages were below 0.2%.

Now we can compare the observations gathered from the simulation results with the data in Sect. 2.2; in particular, we are interested in testing observations 1, 2 and 3 of this section.

Appendix E: Model Validation with Field Data

The “validation” used here for the model is a complex process found in Augusiaka et al. (2014), in which a unifying terminology has been proposed. We focused primarily on model output validation of the qualitative predictions presented in Sect. 3.2. The parameters of the model were estimated from the literature, and no fine-tuning is necessary to observe the patterns predicted. Therefore, the dataset of 32 different sites can be considered independent of model calibration, making it adequate for validation.

In Table 4 we present the classification of the relations between the species as a $11\times 11$ binary matrix, $M_\mathrm{C}$ with 1 representing pertinence to class C and 0 non-pertinence. Since the classes are disjoint, one matrix is enough to describe both classes.

Table 4 List of Myrmica species used to test the patterns

Using the classification, as in Table 4, we have two categories of interspecific competition, C, NC. For each category, if we assign a corresponding coefficient of interspecific competition $c_\mathrm{c}$ and $c_\mathrm{nc}$, the coefficients would obey the relation: $c_\mathrm{c}>c_\mathrm{nc}$, as in Fig. 11. Here $c_\mathrm{c}>c_\mathrm{nc}$ implies that interspecific competition in the class C is stronger than in class NC.

Fig. 11

Relation between the coefficients of interspecific competition for the classes C and NC

From the ordering of the coefficients, Fig. 10 suggests that the correlations between the theoretical primary host abundance ($T_\mathrm{H}$) and the theoretical parasite distribution ($T_\mathrm{P}$) should decrease as c increases. Given the classes C and NC and the expected relations between the coefficients $c_\mathrm{c}$ and $c_\mathrm{nc}$, we should expect the slope of the linear regression between the empirical variables, $E_\mathrm{H}$ and $E_\mathrm{P}$, to increase when comparing the class C with NC. Such are the results presented in Sect. 3.2.