Abstract
Parasite aggregation, a recurring pattern in macroparasite infections, is considered one of the “laws” in parasite ecology. Few hosts have a large number of parasites while most hosts have a low number of parasites. This pattern has been widely studied using phenomenological models, by using the negative binomial distribution. However, to infer the mechanisms of aggregation, a mechanistic model is essential. Here we formulate such a mechanistic model of parasite aggregation in hosts without initially assuming a negative binomial distribution. Our results show that a simple model of parasite accumulation still results in an aggregated pattern, as shown by the derived mean and variance of the parasite distribution. By incorporating the derived mean and variance to the host-parasite interaction, we can predict how aggregation affects the population dynamics of the hosts and parasites through time. Thus, our results can directly be applied to observed data as well as can be utilised in designing statistical sampling procedures. Overall, we have shown how a plausible mechanistic process can result in the often observed phenomenon of parasite aggregation occurring in numerous ecological scenarios.
Key Findings
Parasite aggregation is considered one of the “laws” in parasite ecology – few hosts harbouring a large number of parasites.
While examples abound, there is lack of mechanistic models available to explain the phenomenon
Taking a bottom up approach we construct a simple model of host-parasite population dynamics which naturally results in parasite aggregation – negative binomial distribution of parasites in the host population
While providing a plausible mechanism our model can be readily deployed in field work when designing sampling methodology or analysis of available data.
1 Introduction
Parasites are ubiquitous (Zimmer, 2001). Yet, for parasitologists, sampling can be a hard problem (Cundil and Alexander, 2015; Hollingsworth et al., 2015). This is due to how the parasites are distributed among hosts (Rozsa et al., 2000). While most hosts harbour very few parasites, only a few individuals are hosts to a large number of parasites (Crofton, 1971). Normal distribution is not appropriate to represent such parasite distribution in a host community. This phenomenon, termed as parasite aggregation, is perhaps one of the few “laws” in biology because it is a recurring pattern in nature and finding exceptions to this pattern is rare (Gourbiére et al., 2015; Poulin, 2007; Shaw and Dobson, 1995). The pattern is specifically observed in macroparasite infections, which include diseases brought about by helminths and arthropods (Poulin, 2007; Shaw and Dobson, 1995). The distribution pattern of macroparasites in host populations is an important factor in addressing challenges in investigating disease transmission, such as in the case of human onchocerciasis and schistosomiasis (Basanez and Boussinesq, 1999; Churcher et al., 2005; Guilhem et al., 2012). Aggregation can also affect co-infection by various parasites, parasite-driven evolutionary pressures, and stability of host-parasite communities (Morrill and Forbes, 2012, 2016; Morrill et al., 2017; Wilson et al., 2001). Thus, the study of parasite aggregation addresses a fundamental issue in ecology.
The distribution of hosts with different numbers of parasites can be well captured by the negative binomial (Crofton, 1971; Fisher, 1941; Wilson et al., 1996). Given this fact, a number of theoretical models about host-parasite dynamics implicitly assume the negative binomial distribution, mainly based on a phenomenological (statistical) modeling framework (Adler and Kretzschmar, 1992; Anderson, 1978; Gourbiére et al., 2015; Shaw and Dobson, 1995). The phenomenological principle is based on the observation that - (i) a distribution is Poisson if showing random behavior with equal mean and variance, (ii) binomial if underdispersed with greater mean than variance, or (iii) negative binomial if overdisperesed with greater variance than the mean (Anderson and Gordon, 1982; Wilson et al., 2001). Overdispersion is observed in host-parasite interaction such as between cattle (Bos taurus) and warble fly (Hypoderma bovis), and between Nile tilapia (Oreochromis niloticus) and copepod Ergasilus philippinensis (Lopez, 2001; Wilson et al., 2001). The negative binomial distribution is hypothesized to arise due to the heterogeneity in the characteristics of the hosts and parasites, and variation in the environment. Heterogeneous exposures, infection rates and susceptibility of host individuals are observed to produce aggregated distributions of parasites (Galvani, 2003; Wilber et al., 2017; Wilson et al., 2001). However, is heterogeneity in the characteristics of the hosts and parasites a necessary condition for aggregation? Using a mechanistic model, we show that a regular but non-extreme parasite accumulation can lead to aggregation. This proves that even in regular systems (i.e., with homogeneous proportion of parasite accumulation), aggregation is widely possible.
Traditionally, in arriving at the desired model for host-macroparasite interaction, phenomenological modelling is assumed (Anderson, 1978; Anderson and May, 1978; Crofton, 1971; Rosà and Pugliese, 2002; Yakob et al., 2014). Phenomenological models are based on data gathered, but the mechanisms underlying the phenomenon are generally hidden. There is a need for descriptive and predictive models that assume the underlying mechanistic processes of parasite dynamics, which can be used in formulating disease control programs (Hollingsworth et al., 2015). The goal of this study is to mechanistically illustrate the interaction between hosts and macroparasites without assuming a negative binomial distribution. Our approach considers parasite accumulation without direct reproduction in host individuals, which is a consequence of the complex life history of macroparasites (Auld and Tinsley, 2015; McCallum et al., 2017; Viney and Cable, 2011). For example, the Nile tilapia fish are infected by the acanthocephalans parasites via the ingestion of infected zooplankton. The parasites grow, mate and lay eggs inside the fish, and then eggs are expelled in the lake through faecal excretions. The parasites in the excretions attach to the zooplanktons, and the cycle of infection through foraging continues (de la Cruz et al., 2013; Paller et al., 2016).
Recently, various attempts have been made to model the accumulation and aggregation of parasites mechanistically. The stratified worm burden, which is based on a chain of infected compartments as in Susceptible-Infected (S-I) modeling framework, was used to model schistosomiasis infections (Gurarie et al., 2010, 2015, 2016). This model can be well simulated numerically for specific cases, but possibly difficult to implement analytic studies to find general mathematical conclusions (Gurarie et al., 2010). Moreover, the Poisson-Gamma Mixture Model has been proposed to model aggregation based on parasite accumulation (Calabrese et al., 2011). The negative binomial distribution arises naturally from a Poisson-Gamma process; however, is it possible to derive a mechanistic model of aggregation without initially assuming a distribution related to the negative binomial? In another study, a mechanistic model has been proposed based on the random variation in the exposure of hosts to parasites and the infection success of parasites (Gourbiére et al., 2015). This model anchors its derivation and conclusion on the common assumption that heterogeneity in individual hosts leads to parasite aggregation. Moreover, in most models of macroparasite infections, the rate of disease acquisition by susceptible individuals (known as the “force of infection” β) is assumed as a parameter. However, identifying the value of β can be difficult or infeasible even in the availability of data (McCallum et al., 2017). In our proposed model, this challenge can be addressed since our assumed infection parameter, denoted by P, can be directly estimated from parasite count data gathered from host samples in empirical studies. Our model is simple and tractable, and can be incorporated as part of classical modeling frameworks (e.g., host-parasite interaction with logistic growth) (Edelstein-Keshet, 2005).
2 Model
The core concept of our model is captured in Fig. 1. The total host population has a density of X. With probability P0, the hosts are parasite-free and the density of parasite-free hosts is then X0 = P0X. In a similar fashion, the density of hosts which have at least one parasite is X1 = P1X. The total host population is there-fore equivalent to X = X0 + X1 = P0X + P1X. Moreoever, the class of individuals Xi which have at least i > 1 number of parasites is assumed as a subset of X1 (i.e., Xn+1 ⊆ Xn ⊆ … ⊆ X2 ⊆ X1), and can be modeled as,
From the above set of equations, we can derive the distribution of living hosts (all except the n+1 compartment) with different parasite load with respect to the total host population. Fig. 1 represents the classification of the states of the hosts depending on their parasite load. The compartment associated with Xi, i ≥ 1 contains the hosts infected by at least i number of parasites. The parameter Pi+1 is the probability that a host in Xi acquires an additional parasite. Consequently, the difference Xi −Xi+1, i ≥ 1 is the population density of hosts with exactly i number of parasites. The population density of living hosts (with maximum tolerable parasite load n) is then Xn − Xn+1.
Thus the death of hosts due to the parasite is captured by the transition into the Xn+1 state given by , the number of dead hosts (last compartment in Eqs. (1)).
Our model, described as such, does not follow the classical input-output modeling framework (e.g., stratified worm burden model which is an extension of the standard S-I epidemiological models). Such a classical modeling framework could be intractable when modeling aggregation (Gurarie et al., 2010). The number of variables and equations in the stratified worm burden model can increase as the number of states (compartments) increases. Here, we model the states in the compartment diagram using proportions so we can easily analyze the distribution of parasites using probabilities. The dynamics of parasite infection can then be summarized using the properties of the derived distributions, such as the mean and variance. Also, in input-output models, the transfer from one state (e.g., susceptible) to another (e.g., state with 3 parasites) needs to pass through intermediate diseased states (e.g., states with 1 and 2 parasites, respectively). In our model, a host can acquire more than 1 parasite, and this can be modeled by adjusting the parameter Pi. This is consistent with the experimental approaches and comparable to the data gathered. For example, the proportion Pi is directly computable from parasite load data from sampled hosts as compared to computing the force of infection β in S-I models (Gandon and Day, 2009).
The quantities relating to the host and parasite proportions, Xi, Pi and n can be dynamic (e.g., may change over time). Here, we assume that Xi changes following a logistic growth with parasitism, and Pi is a function of parasite population Y (hence, also a function of time). This allows us to connect our model to experimental data. The parameter values (e.g., Pi) can be determined from the samples gathered at a certain time instant, and the pattern of the temporal evolution of the parameter values can be inferred from time series data.
Depending on the exact transmission mode of the parasite, Pi can have different functional forms. A basic assumption would be that Pi is a function of parasite encounter and transmission rates. For example, we could have for all i ≠ 0. Here, the total ecological carrying capacity for the parasite population is assumed to be nK + c, where K is the host carrying capacity and n is the maximum number of parasites that a living host can harbour without dying. The parameter c is the quantitative representation of the environment where the parasites can survive outside the hosts. Thus we can interpret as the parasite encounter probability with p being the parasite transmission probability. We focus on cases where parasites highly depend on the hosts to survive, and without loss of generality, we assume a small value c = 1. Assuming homogeneity in parasite transmission, p is kept constant. Moreover, the host basal growth rate is set to a constant rH, assuming that macroparasites do not affect reproduction of hosts.
For the parasite population, the total parasite density in host population is where Yi = i(Xi − Xi−1), i ≥ 1. The distribution of parasites is according to the following:
Based on the set of Eqs. (2), the parasite population density in host population can be written as . Here, N is the random variable representing the parasite load in a living host, and is the approximate mean of N.
The carrying capacity of the host population is assumed to be equal to K. The death rate of the hosts is assumed to be , which is derived from the equation representing Xn+1. Moreover, the per capita parasite reproduction rate is assumed to be rp. Together with the carrying capacity for the parasites (nK + 1) the population dynamics between the hosts and parasites can be modeled assuming logistic growth as follows (refer to Table SI.1 for the description of the state variables and parameters):
3 Results
The distribution of the parasites in the living hosts is represented by . The general expression for this distributions as discussed above is given by, where A is derived from the derivative of a geometric series (see SI.1.2). If the parasite transmission (for i ≠ 0), then the distribution of the parasites in the living hosts has since all hosts that are harbouring at least n + 1 number of parasites are dead. Now, supposing, , we have, as the mean, and the variance, , derived in the SI.1.3.
A negative binomial distribution describing the probability distribution of the number of successes before the m-th failure, where ρ is the probability of success can be written as NB(m, ρ). The mean and variance of NB(m, ρ) are and , respectively. From Eq. (6), is equivalent to the mean of a negative binomial distribution with m = nPn+1 − (n + 1)Pn + 1 and ρ = P. For the non-truncated negative binomial distribution, we consider n → ∞ (Crofton, 1971; Shonkwiler, 2016). In the next section (3.1), we discuss that as and respectively converge to the mean and variance of a geometric distribution (NB(1, ρ)).
3.1 Constant host-parasite encounter probability
Suppose the parasite encounter probability is fixed. This implies that whatever the values of Y and is always constant. We can also interpret P as the constant geometric mean of the parasite acquisition probabilities Pi, i ≥ 1.
For a large n and and approximate the mean and variance of N ∈ {0, 1, 2, …, n}, respectively. E[N] can be interpreted as the average number of parasites in a host with variance V ar[N]. From Eq. (6) as is equivalent to the mean of a negative binomial distribution with m = 1 and ρ = P, which characterizes the mean of a geometric distribution. Let us denote this mean and variance as
The variance-to-mean ratio is , which increases as P increases (Table SI.2).
The implication of this result is that even if the host can sustain a large number of parasites (n → ∞) for P < 1, we have finite mean and variance at the population level. Also, the variance-to-mean ratio is greater than 1 characterizing an over-dispersed distribution of parasite load in the host population. For example, E[∞]0.5 implies that the average number of parasites in a host is 1 with variance V ar[∞]0.5 = 2. E[∞]0.9 implies that the average number of parasites in a host is 9 with variance V ar[∞]0.9 = 90.
As stated earlier, E[∞]P is clearly an approximation. The error due to the approximation can be estimated and is shown in Fig. 2A. For a wide range of parameter values of the maximum tolerable parasite load of the host (n) and of the probability of parasite acquisition by a host (P) we see that the mean of the geometric distribution is a good estimate for . Fig. 2A further illustrates that as n approaches infinity, the error becomes smaller. While, a small value for n and a higher P produce higher error, this could also be an improbable scenario in nature. The condition where the probability of acquiring more parasites is high and the number of tolerable parasites is low, would result in higher infection rates and potentially lead to the extinction of the hosts. This is illustrated in Fig. 3 where high parasite-driven host death drives the host population extinct .
The same is true with the approximation of by V ar[∞]P. Fig 2B shows that as n → ∞, the error becomes smaller. This illustrates that the variance of the geometric distribution is a good estimate for (Fig. 3).
In the supplementary information Figs. SI.1 and SI.2, we present examples of parasite load distribution in host population with differing values of P. An intermediate value of P results in aggregated distribution. However, the distribution becomes more negatively skewed as P increases which shows high host mortality due to harbouring high parasite load. This is the reason why as P increases, the errors in approximating and using the geometric distribution also increase.
3.1.1 Population dynamics
We now analyze in detail the population dynamics between the hosts and parasites (Eqs. (3) and (4)). Since the probability of parasite acquisition by a host P is constant, we can decouple the host-parasite dynamics. Analyzing first the host population (Eq. (3)), we have a logistic growth function with death or harvesting term (D = Pn+1X) (Clark, 2010). In Fig. 3, the blue curve represents the logistic growth function G. The red line represents the death function D. The intersection of the two curve is an equilibrium.
There are two equilibrium points if rH > Pn+1, where one is unstable and the other is stable (Fig. 3A). If rH ≤ Pn+1, the zero equilibrium point is stable and the only steady state of the dynamics (Fig. 3B), implying an eventual extinction of the host population. For the parasites to exploit the maximum growth rate of the hosts (rH K/4) (Fig. 3), the parasite-driven host death rate should be Pn+1 = rH /2.
Now, analyzing the parasite population (Eq. (4)), there are two possible equilibrium states: the parasite can go extinct, which happens if X = 0 or , and a stable coexistence of hosts and parasites at where X* is the host equilibrium. The equilibrium state is unstable, and is stable. The condition for only happens if P = 0 (proof in SI.1.4).
Over time the total parasite population density in living host population converges to . The expected number of parasites in the hosts then tends to n (parasites in the environment represented by c are not included since they are outside the living hosts). One might think that this limiting case may not be the situation if parasite distribution in living hosts is aggregated. If aggregation affects the carrying capacity of the parasite population, the parameter n in the denominator nX + 1 in Eq. 4 can be replaced by where σ represents the contribution of the variance to the average number of parasites in a host. Hence, if the distribution associated with parasite aggregation is considered, as time t → ∞. In the next section, we investigate the case when P is not constant through time.
3.2 Variable host-parasite encounter probability
Until now we assumed that the parasite encounter probability P as a constant. However, (Eqs. (3) and (4)) is a function of the dynamic variable Y, the parasite density. Given the dynamics of the parasite, we can have three possible equilibrium points. Two of the equilibrium points are trivial, one where no host and parasite exist (X* = 0, Y * = Y0), and another at the carrying capacity of the hosts (X* = K, Y * = Y0 = 0) with representing the disease-free state.
The third equilibrium point posits a coexistence of hosts and parasites and can be derived from
This leads to which we can analyze by investigating the intersection of the curves formed by the left and right hand sides of this equation. Suppose this intersection is X* = α > 0 (Fig. 4). The equilibrium point is then (X* = α, Y * = nα + 1) where α satisfies Eq. (10). This equilibrium point is stable (Fig. 4).
If parasite aggregation affects the carrying capacity of the parasite population, we can replace Y * in Eq. 9 by an implicit equation . Since , the new equilibrium state, if it exists, is X* = γ > α, where α is the equilibrium state if Y * = nX* + 1 (Fig. 4). This means that aggregation limits the parasitism-driven stress affecting the host population. Sample numerical simulations of host-parasite population dynamics are shown in Fig. 5 with varying values of σ. The value of converges to n if the value of σ is increased. In the example, notice that an aggregated parasite distribution with σ = 2 increases the parasite population rapidly but without causing too much harm to the host population (Fig. 5).
4 Discussion
Negative binomial distribution is commonly used in modeling macroparasite infections (Pennycuick, 1971; Wegner et al., 2008). Here, we propose a mechanistic model of parasite aggregation without initially assuming a statistical distribution. Our results show that the emergent values of the mean and variance of the macroparasite distribution indeed denote a negative binomial. We have shown that accumulation of macroparasites (e.g., through foraging) is sufficient for aggregation to arise under a wide range of conditions (Fig. 2). The complex life cycle of parasites, a relatively large maximum tolerable parasite load n, and relatively moderate parasite acquisition probability Pi are important factors in parasite aggregation. This means that parasites have the opportunity to reproduce through infecting hosts but without killing many hosts. It would be rare in nature to find hosts having low n with parasites having high Pi resulting in non-aggregated parasite distribution since these hosts (as well as the parasites, if specific) are expected to be extinct.
Our model design can be used to predict what conditions result in over-dispersed (aggregated), under-dispersed, and random parasite distributions by investigating the values of the parasite acquisition parameter Pi. Over-dispersion is observed when the variance of the parasite load in hosts is higher than the mean, while under-dispersion has a mean higher than its variance. Random pattern arises if mean equals the variance. This prediction is valuable in parasitology when performing empirical studies, especially when designing statistical sampling procedures. If the parasites are aggregated in the host population, then a large number of samples is expected to be needed to select those hosts with high parasite load at the tail of the distribution (Shvydka et al., 2018).
The main advantage of our model compared to the classical input-output modelling framework is its simplicity without losing important biological details, such as we can model host-parasite interaction using minimal models but still aggregation is considered. The designed model framework has few variables since the effect of parasite distribution can be summarised using its moments (mean and variance). The framework supports traditional population dynamic models, such as the logistic host-parasite interaction model, which are amenable to numerical and analytic mathematical investigations. Remarkably, our assumed parameters, especially Pi, can be directly calculated from available empirical data. Moreover, our model is more general than the stratified worm burden in terms of parasite acquisition (Gurarie et al., 2010). In stratified worm burden, a host can only acquire one parasite at a time. In our model, hosts can acquire more than one parasite since compartments associated with Xi are defined as hosts with “at least” (not “exactly”) i parasite load, a scenario impossible in the stratified worm burden model.
There are multiple indices that are proposed to measure parasite aggregation. One example is the negative binomial parameter k, which is equivalent to m in NB(m, ρ). This is defined by the equation V ar[N] = E[N] + E2[N]/k. As k decreases to zero (e.g, less than 1 but positive), the parasite distribution is said to become more aggregated. This low k can also infer heterogeneity in infection factors (Bolker, 2008). If k approaches infinity, the Poisson distribution results. Increases in k are used to indicate a movement toward “randomness” (Bolker, 2008; Young and Young, 1990). In the case of parasite population following a geometric distribution, k = 1. Based on our results, we have identified parameters that affect the aggregation index k, most notably are the parameters Pi and n. This implies that Pi can be used as an alternative measure of aggregation that biologists can use in studying patterns of macroparasite distribution in hosts.
Given a set of parasite counts gathered from host samples, one can calculate the estimates for and . If we want to calculate the expected value of N ∈ {0, 1, 2, …, n} without assuming a large n, we can simply normalize the probabilities associated with N. That is, . Similarly, we can do this normalization to obtain the variance of N, V ar[N]. The variables and parameters Xi, Yi, Pi, E[N] and V ar[N] can be functions of time, and a time series analysis to study the temporal pattern of these variables and parameters can be implemented. If Pi’s, the host-parasite encounter probabilities, are statistically equal, we can assume a fixed P for each unit of time by taking the geometric mean of Pi (i ≥ 1), that is, .
The mortality rate due to parasitism in Eq. (3) can be modified to include the cases where a fraction of the hosts with i < n number of parasites could also die due to infection. This rate can be formulated as bX where b is as follows:
The parameter ωi is the fraction of Xi − Xi+1 killed by the parasites. Our conclusions from the qualitative analysis, especially given a fixed P = Pi for i ≥ 1, still remain true since we can suppose the mortality rate b as the slope of the parasite-driven host death in Fig. 3.
Here, we have shown the resulting distribution of parasites in the host population using homogeneous parasite acquisition probability. If the values of the acquisition probabilities become heterogeneous, such as if P1 < P2 < … < Pn, P1 > P2 > … > Pn or Pi’s are completely arbitrary, then our derived formulas to estimate E[N] and V ar[N] are not applicable (e.g., Eq. (6)). However, our qualitative analysis to investigate the dynamics of Eqs. (3) and (4) could still hold and a numerical study would then be the feasible way forward. To find appropriate formulae (if data are not available) for and would then be a challenge for the future.
Various future studies can stem from our model design. One can include the existence of alternative or intermediate hosts and vectors, especially that neglected tropical diseases are commonly due to vector-borne macroparasite infections (Hollingsworth et al., 2015). Our model can also be extended to include how spatial aspects and different treatment strategies affect the population dynamics of the hosts and parasites, and to infer the reasons why the value of k is dynamic as observed in empirical studies (Boag et al., 2001; Crompton et al., 1984; Pennycuick, 1971; Scott, 1987). Inclusion of multiple parasites (Hafer and Milinski, 2015), different models of host-parasite interactions in food webs (Flor, 1956), or the explicit inclusion of population dynamics together with host-parasite co-evolution (Gokhale et al., 2013; Rabajante et al., 2016; Song et al., 2015) are possible directions. Testing if such complexities retain the observed phenomenon of parasite aggregation would be a true test of the “law”.
6 Financial Support
JFR and ELA are supported by the Institute of Mathematical Sciences and Physics, University of the Philippines Los Baños. CSG is supported by the Max Planck Society.
7 Declarations
The authors declare that there are no conflicts of interest. The funding sources had no involvement in the study design, in the analysis and interpretation of results, in the writing of the manuscript and in the decision to submit the article for publication.
8 Author Contributions
JFR conceptualized the problem. JFR and CSG designed the model. JFR and ELA implemented the simulations. All authors contributed in the analysis of the model, interpretation of results, writing of the manuscript and preparation of figures.
9 Data Availability
SI.1 Supplementary Information
SI.1.1 List of state variable and parameters
Since we have the mean of the random variable N + 1 is
The random variable N + 1 ∈ {0, 1, 2, …, n, w} (where w ≥ n + 1) represents the parasite load in living and parasitism-driven dead hosts. For simplicity, we let w = n+1 since hosts with equal or more than n + 1 parasites are dead due to parasitism. Let us define , which is the approximate mean of N (where N ∈ {0, 1, 2, …, n}) is as discussed in the main text.
SI.1.2 Derivation of A
We assume that the parasite acquisition probability is . Using the geometric series, we know that,
Then taking the derivative of this geometric series results in
The expression for A is:
SI.1.3 Explicit formula for the variance
Let us define where V ar[N + 1] = E[(N + 1)2] − E[N + 1]2 is the variance of the random variable N + 1.
With the same assumption as above for P, The expression for E[(N + 1)2] is
The expression for B can be derived using the geometric series. We know that
Multiplying both sides by P, we have
Taking the derivative of the left- and right-hand sides, we arrive at the following expression for B:
Hence, the explicit formula for is
SI.1.4 Proof for a claim in Section 3.1
Here is the proof that when only happens if P = 0. Suppose P < 1: implies P = 0 or nPn+1 − (n + 1) Pn + 1 = 0. If nPn+1 − (n + 1) Pn + 1 = 0 then
Note that represents the geometric series:
However, is the arithmetic average of {Pn−1, Pn−2, …, P1, P0}. This is a contradiction since Pn ∉ {Pn−1, Pn−2, …, P1, P0}.
Suppose P = 1: Note that is fixed. If P = 1, then or . But and p < ∞, a contradiction.
Hence, only if P = 0.
SI.1.5 Histograms given different values of P, n = 10
5 Acknowledgements
JFR appreciates the time spent at the Max Planck Institute for Evolutionary Biology developing the project.