Broadly inflicted stressors can cause ecosystem thinning

Abstract

Many anthropogenic stressors broadly inflict mortality or reduce fecundity, including habitat destruction, pollution, climate change, invasive species, and multispecies harvesting. Here, we show—in four analytical models of interspecies competition—that broadly inflicted stressors disproportionately cause competitive exclusions within groups of ecologically similar species. As a result, we predict that ecosystems become progressively thinner—that is, they have progressively less functional redundancy—as broadly inflicted stressors become progressively more intense. This may negatively affect the temporal stability of ecosystem functions, but it also buffers ecosystem productivity against stress by favoring species less sensitive to the stressors. Our main result follows from the weak limiting similarity principle: species with more similar ecological niches compete more strongly, and their coexistence can be upset by smaller perturbations. We show that stressors can cause indirect competitive exclusions at much lower stressor intensity than needed to directly cause species extinction, consistent with the finding of empirical studies that species interactions are often the proximal drivers of local extinctions. The excluded species are more sensitive to the stressor relative to their ecologically similar competitors. Moreover, broadly inflicted stressors may cause hydra effects—where higher stressor intensity results in higher abundance for a species with lower sensitivity to the stressor than its competitors. Correlations between stressor impacts and ecological niches reduce the potential for indirect competitive exclusions, but they consequently also reduce the buffering effect of ecosystem thinning on ecosystem productivity. Our findings suggest that ecosystems experiencing stress may continue to provision ecosystem services but lose functional redundancy and stability.

Appendix. Weak limiting similarity and broadly inflicted stress in mechanistic models

Our main results follow from the weak limiting similarity principle: the more ecologically similar two species are, the more competitively similar they must be to coexist and the smaller the perturbation needed to upset coexistence. In the context of broadly inflicted stress, weak limiting similarity implies that—barring the special case where the stressor identically impacts species’ competitive abilities—lower-intensity stress is needed, on average, to upset the coexistence between more ecologically similar species. We demonstrate this principle in a Lotka-Volterra-type competition model in the main text. Below, we show that principle is robust to each of three qualitatively different mechanistic competition models (Fig. 3) (see Table 1 for all parameter definitions).

Model 1: resource competition

Our first model is based on Tilman’s (1980) model of two species competing for two limiting resources (Fig. 3a). The abundances of these resources (1 and 2) are denoted R₁ and R₂. We assume that the rates of change in abundance of species i and resource j are given by

$$ \frac{d{N}_i}{N_i dt}={r}_i\left[{f}_i\left({R}_1,{R}_2\right)-{m}_i\left(1+{v}_iE\right)\right] $$

(14)

$$ \frac{d{R}_j}{dt}={g}_j\left({R}_j\right)-{\sum}_{i=1}^2{N}_i{r}_i{f}_i\left({R}_1,{R}_2\right){h}_{ij}\left({R}_1,{R}_2\right). $$

(15)

Here, g_j(.) is a function describing the growth rate of resource j in the absence of consumption by the species, f_i(.) describes the growth rate of species i as a function of resources, m_i is the natural mortality rate of species i, and h_ij(.) describes the conversion rate of resource j into species i. To be consistent with the other models in our analysis, we add the parameter r_i (which Tilman 1980 does not use) to represent the scale of turnover of species i. Thus, f_i and m_i in our model are normalized measures. We also normalize sensitivity (v_i) by m_i. This allows v_i (sensitivity) to have a similar interpretation as it does in the main text. The parameter r_i does not quite represent the maximum per-capita growth rate here—as it does in the main text—but serves an analogous function, by measuring the scale of turnover, as we will see below.

For simplicity, we focus our analysis on the case of essential resources—in which the resource ratio required by each species provides a univariate measure of its niche, which we denote Z_i for species i. However, we conjecture that our main result should also hold with most other types of resources analyzed in Tilman 1980. We define r_i such that f_i can be expressed as

$$ {f}_i\left({R}_1,{R}_2\right)=\min \left\{{R}_1,{Z}_i{R}_2\right\}. $$

(16)

The fact that the resources are essential is captured by the fact that species i grows according to the smaller value of R₁ and Z_iR₂; thus, Z_i represents the amount of resource 1 relative to resource 2 that species i needs to grow (the resource ratio), which is the key measure of the niche in this model. For instance, a species’ niche in this type of model might be how much light or phosphorus it needs relative to water or nitrogen (e.g., Titman 1976).

With these assumptions, the minimum concentrations of the resources allowing the persistence of species i (i.e., allowing $\frac{d{N}_i}{dt}\ge 0$), denoted R_1i^* and R_2i^* for species i, are

$$ {R_{1i}}^{\ast }={m}_i\left(1+{v}_iE\right) $$

(17)

$$ {R_{2i}}^{\ast }=\frac{m_i}{Z_i}\left(1+{v}_iE\right). $$

(18)

Tilman (1980) showed that a necessary condition for the coexistence of the two species in this model is that if R_1i^* < R_1j^*, R_2i^* > R_2j^*. In other words, the two species can only coexist if whichever is the superior competitor with respect to resource 1 (i.e., with a lower R_1i^*) is the inferior competitor with respect to resource 2 (higher R_2i^*) (Fig. 3b). If one species was a better competitor with respect to both resources, it would outcompete the other. With no stressor (E = 0), this necessary condition for coexistence (i.e., if R_1i^* < R_1j^* then R_2i^* > R_2j^*) amounts to

$$ 1<\frac{m_j}{m_i}<\frac{Z_j}{Z_i},\mathrm{when}\ {m}_j>{m}_i., $$

(19)

With the addition of the stressor (E > 0), the analogous necessary condition for coexistence is

$$ 1<\frac{m_j\left(1+{v}_jE\right)}{m_i\left(1+{v}_iE\right)}<\frac{Z_j}{Z_i},w\mathrm{hen}\ {m}_j>{m}_i. $$

(20)

Equations (19) and (20) are analogous to Eqs. (3a) and (3b), from the main text (Fig. 3c). The more ecologically similar the two species (i.e., the closer Z_i/Z_j is to 1), the smaller the range of relative mortality rates (m_j/m_i) allowing coexistence. If v_i = v_j, increasing the stressor intensity does not perturb coexistence, but if v_i > v_j, species i will become less competitive (i.e., $\frac{m_j\left(1+{v}_jE\right)}{m_i\left(1+{v}_iE\right)}$ will decrease) as E increases. The limit of $\frac{1+{v}_jE}{1+{v}_iE}$, as E approaches infinity, is $\frac{v_j}{v_i}$, so even with v_i > v_j, necessary coexistence Eq. (20) cannot be disrupted by stress if the difference in sensitivity is too small relative to the difference in mortality rate: specifically, if $\frac{v_i}{v_j}<\frac{m_j}{m_i}$ when m_j > m_i, or equivalently, if $\frac{v_i}{v_j}<\frac{m_i}{m_j}$ when m_i > m_j. However, m_i/m_j approaches 1 as Z_i/Z_j approaches 1, among pairs of coexisting species [by Eq. (20)], making the minimum sensitivity difference permitting competitive exclusion [by Eq. (20)] vanish as the niches approach total similarity (i.e., as Z_i/Z_j approaches 1). If sensitivity differences are sufficiently large to allow competitive exclusion by increasing stress (E) according to Eq. (20), the stressor intensity causing competitive exclusion of species i, when v_i > v_j, is bounded by

$$ 0<{E}_{\mathrm{c}}\le \frac{\frac{Z_j}{Z_i}-1}{v_j\left(\frac{v_i}{v_j}-\frac{Z_j}{Z_i}\right)},\mathrm{when}\ {m}_j>{m}_i\left(\mathrm{implying}\ {Z}_j>{Z}_i\right) $$

(21a)

$$ 0<{E}_{\mathrm{c}}\le \frac{\frac{Z_i}{Z_j}-1}{v_j\left(\frac{v_i}{v_j}-\frac{Z_i}{Z_j}\right)},\mathrm{when}\ {m}_i>{m}_j\left(\mathrm{implying}\ {Z}_i>{Z}_j\right). $$

(21b)

This is analogous to Eq. (4) in the main text and has an analogous interpretation—the upper bound approaches zero as the niches approach total similarity (i.e., as Z_i/Z_j approaches 1).

Model 2: apparent and exploitative competition

Our second model is based on Holt et al.’s (1994) model of apparent and exploitative competition (Fig. 3d). Here, the two competing species compete directly for a single resource, with abundance R. They also indirectly compete via sharing a predator, having abundance P. In the absence of stressors (E = 0), the system is assumed to be closed, such that there is a constant supply (s) of total resource in the system, which transfers between forms (resource, competitor, predator). The dynamics of the system, without the stressor, are given by

$$ \frac{dP}{Pdt}={\sum}_{i=1}^2{a}_i{b}_i{N}_i-d $$

(22)

$$ \frac{d{N}_i}{N_i dt}={r}_i\left[R-{R_i}^{\ast }-{\alpha}_iP\right],\mathrm{where}\kern0.5em {r}_i={a_i}^{\prime }{b_i}^{\prime },{R_i}^{\ast }=\frac{{d_i}^{\prime }}{r_i}\kern0.28em \mathrm{and}\kern0.5em {\alpha}_i=\frac{a_i}{r_i} $$

(23)

$$ R=s-{\sum}_{i=1}^2\frac{N_i}{{b_i}^{\prime }}-\frac{P}{b_P}. $$

(24)

Here, a_i is the consumption rate of competitor i by predators; b_i is the conversion rate of competitor i consumed into predator biomass; a_i^′b_i^′ is the analogous consumption and conversion rates, from resources into competitor i; d and d_i^′ are the natural death rates, of predators and competitor i, respectively; and b_P is the effective conversion rate of resources into predators. The other variables defined in Eq. (23) are useful permutations of these. For instance, R_i^* is the minimum resource abundance on which species i can survive if there are no predators or stressors (Holt et al. 1994).

Following Holt et al. (1994), we denote the equilibrium resource and predator abundances that would occur with no stressor and only competitor i, R_i^∗∗, and P_i^∗∗, respectively. These, together with R_i^∗, determine species i’s niche, and they can be derived from Eqs. (22) to (24)

$$ {R_i}^{\ast \ast }={R_i}^{\ast }+{\alpha}_i{P_i}^{\ast \ast }. $$

(25)

$$ {P_i}^{\ast \ast }=\frac{b_P\left(s{a}_i{b}_i{b_i}^{\prime }-d-{R_i}^{\ast }{a}_i{b}_i{b_i}^{\prime}\right)}{a_i{b}_i{b_i}^{\prime}\left({\alpha}_i{b}_P+1\right)}. $$

(26)

Stable coexistence requires both species to be able to increase when rare. For species i, this means that $\frac{d{N}_i}{dt}>0$ [from Eq. (23)] when N_i ≈ 0, R = R_j^∗∗, and P = P_j^∗∗. Without the stressor, this condition is equivalent to

$$ {R_j}^{\ast \ast }-{R_i}^{\ast \ast }>{\alpha}_i\left({P_j}^{\ast \ast }-{P_i}^{\ast \ast}\right). $$

(27)

Re-arranging Eq. (25), we see that

$$ {\alpha}_i=\frac{{R_i}^{\ast \ast }-{R_i}^{\ast }}{{P_i}^{\ast \ast }}. $$

(28)

Thus, the parameter α_i is the key measure of the niche for our purposes. It measures the amount of additional resources species i needs, per additional predator. Coexistence without the stressor requires an implicit tradeoff between resource use and predator resistance (Holt et al. 1994) (Fig. 3e). We can see this from inequality Eq. (27), which cannot hold for both species (each having positive α) unless (R_j^∗∗ − R_i^∗∗) and (P_j^∗∗ − P_i^∗∗) have the same sign—meaning that whichever species can survive on fewer resources (lower R^**) must also be able to tolerate a lower predator abundance (lower P^**). The parameter α_i effectively measures where, on this tradeoff, species i lies: small α_i means that species i needs relatively few additional resources to cope with additional predators (i.e., it specializes in predator resistance), and large α_i means species i’s resource requirements increase significantly as predators become more abundant, meaning it would have to be a better resource competitor to coexist.

Suppose species j is the inferior resource competitor, but is less susceptible to predation, such that R_j^∗∗ > R_i^∗∗ and P_j^∗∗ > P_i^∗∗. In this case, we can simultaneously express the coexistence criterion Eq. (27) for both species as

$$ {\alpha}_i<\frac{{R_j}^{\ast \ast }-{R_i}^{\ast \ast }}{{P_j}^{\ast \ast }-{P_i}^{\ast \ast }}<{\alpha}_j,\mathrm{when}\;{R_j}^{\ast \ast }>{R_i}^{\ast \ast}\mathrm{and}\ {P_j}^{\ast \ast }>{P_i}^{\ast \ast }. $$

(29)

Coexistence Eq. (29) is analogous to coexistence Eqs. (19) and (3a) from the previous models (Fig. 3f). The more similar the two species’ niches (measured by α_i and α_j) are, the more similar their competitive abilities must be to coexist. As α_i approaches α_j, coexistence requires $\frac{{R_j}^{\ast \ast }-{R_i}^{\ast \ast }}{{P_j}^{\ast \ast }-{P_i}^{\ast \ast }}$ approach 1, and any small perturbation in R_i^∗∗, R_j^∗∗, P_i^∗∗, or P_j^∗∗, caused by a stressor, would disrupt coexistence.

Suppose that stress (having intensity E) negatively affects the per-capita growth rate of the two consumer species (1 and 2) in some way. By increasing mortality on the consumer species, stress would have a positive direct effect on R_i^∗∗ (because species i needs more resources to offset higher mortality), but it will have a negative indirect effect on both R_i^∗∗ and P_i^∗∗ by reducing the efficiency with which resources are passed through the food chain to predators (which lowers R_i^∗∗ by reducing predation pressure). By removing biomass from the system, the stressor would also lower the mass-balance constraint (i.e., $\frac{\partial s}{\partial E}<0$).

For instance, suppose the dynamics of the consumer species were now given by

$$ \frac{d{N}_i}{N_i dt}={r}_i\left[R-{R_i}^{\ast }-{\alpha}_iP-{\gamma}_iE\right]; $$

(30)

where r_iγ_i is the per-capita, per-unit intensity mortality rate inflicted by the stressor on species i. Suppose also that the mass-balance constraint was affected by the stress, such that there was less biomass in the system (e.g., because of harvesting); i.e., s = s(E) and s^′(E) < 0. In this case, the equilibrium resource and predator abundances that would be reached with stress (E > 0) and only species i (no species j), denoted R_i^′ and P_i^′, respectively, are given by

$$ {R_i}^{\prime }={R_i}^{\ast \ast }+{\gamma}_iE-{\alpha}_i\frac{b_P\left[s(0)-s(E)+{\gamma}_iE\right]}{\alpha_i{b}_P+1} $$

(31)

$$ {P_i}^{\prime }={P_i}^{\ast \ast }-\frac{b_P\left[s(0)-s(E)+{\gamma}_iE\right]}{\alpha_i{b}_P+1}. $$

(32)

From Eq. (32), we can see that the term $\frac{b_P\left[s(0)-s(E)+{\gamma}_iE\right]}{\alpha_i{b}_P+1}$ represents the reduction in equilibrium predator abundance (P_i^∗∗ − P_i^′) caused indirectly by the stressor, when species j is not present. Thus, from Eq. (31), the increase in equilibrium resource abundance caused by the stressor (R_i^′ − R_i^∗∗), when species j is not present, is the sum of two impacts: the effect of increasing species i’s mortality (γ_iE), which reduces pressure on the resource, and the effect of reducing predator abundance (${\alpha}_i\frac{b_P\left[s(0)-s(E)+{\gamma}_iE\right]}{\alpha_i{b}_P+1}$), which indirectly increases pressure on the resource.

Thus, defining a sensitivity parameter, v_i, is not as straightforward in this model as in the previous two. Even though the definition of γ_i in Eq. (30) seems analogous to v_i in the previous models, it is not the case where γ_i = γ_j which implies no effect of stress on coexistence. However, the main model insight regarding weak similarity and broadly inflicted stress is the same: There is still a single special case in which stress does not affect coexistence, namely when $\frac{\partial \left({R_j}^{\prime }-{R_i}^{\prime}\right)}{\left({R_j}^{\prime }-{R_i}^{\prime}\right)\partial E}=\frac{\partial \left({P_j}^{\prime }-{P_i}^{\prime}\right)}{\left({P_j}^{\prime }-{P_i}^{\prime}\right)\partial E}$. This case is analogous to the equal-sensitivity cases in the previous two models. If $\frac{\partial \left({R_j}^{\prime }-{R_i}^{\prime}\right)}{\left({R_j}^{\prime }-{R_i}^{\prime}\right)\partial E}>\frac{\partial \left({P_j}^{\prime }-{P_i}^{\prime}\right)}{\left({P_j}^{\prime }-{P_i}^{\prime}\right)\partial E}$, increasing stress perturbs competition to the disadvantage of species j (the predator specialist), and if $\frac{\partial \left({R_j}^{\prime }-{R_i}^{\prime}\right)}{\left({R_j}^{\prime }-{R_i}^{\prime}\right)\partial E}<\frac{\partial \left({P_j}^{\prime }-{P_i}^{\prime}\right)}{\left({P_j}^{\prime }-{P_i}^{\prime}\right)\partial E}$, increasing stress perturbs competition to the disadvantage of species i (the resource specialist). It is also possible to derive bounds, analogous to Eqs. (4), (21a), and (21b), on the stressor intensity, E_c, causing competitive exclusion of the more sensitive competitor—the upper bound approaches zero as α_j approaches α_i. For instance, suppose stress disadvantages species i (as in the examples in the previous models), i.e., $\frac{\partial \left({R_j}^{\prime }-{R_i}^{\prime}\right)}{\left({R_j}^{\prime }-{R_i}^{\prime}\right)\partial E}<\frac{\partial \left({P_j}^{\prime }-{P_i}^{\prime}\right)}{\left({P_j}^{\prime }-{P_i}^{\prime}\right)\partial E}$; stress impacts both consumer species as assumed by Eq. (30); R_j^∗∗ > R_i^∗∗ and P_j^∗∗ > P_i^∗∗; and s(E) = s₀ − s^′E. Then

$$ 0<{E}_{\mathrm{c}}\le \frac{\left({\alpha}_j-{\alpha}_i\right)\left({P_j}^{\ast \ast }-{P_i}^{\ast \ast}\right)}{\left[\left({\alpha}_i-{\alpha}_j\right)\left(\frac{b_P\left[{s}^{\prime }+{\gamma}_i\right]}{\alpha_i{b}_P+1}-\frac{b_P\left[{s}^{\prime }+{\gamma}_j\right]}{\alpha_j{b}_P+1}\right)+{\gamma}_i-{\gamma}_j\right]}. $$

(33)

The upper bound of E_c approaches zero as the niches approach total similarity (i.e., as α_i approaches α_j).

Model 3: competition-colonization tradeoff

Our third and final mechanistic model (Fig. 3g) is based on that of Calcagno et al. (2006). It explores a competition-colonization tradeoff in which differences between species in local competitive ability are cardinal (i.e., numeric and continuous) rather than ordinal [i.e., ranked, but not continuous; differences in competitive ability are ordinal in earlier competition-colonization models such as those of Levins (1969), Tilman (1994), or Kinzig et al. (1999)]. In this model, abundance is measured in terms of the proportion of total sites in the ecosystem that are occupied by each species, p_i for species i. Two species cannot occupy the same space (i.e., p₁ + p₂ ≤ 1). Species i attempts to colonize sites that it does not currently occupy at rate c_i; it is always successful when attempting to colonize empty sites. Species i is exterminated from sites it occupies at rate M_i, by a combination of natural processes and the anthropogenic stressor. We define a standardized mortality measure, m_i, which measures mortality as a fraction of the colonization rate (m_i = M_i/c_i). When species i attempts to colonize a site in which species j is currently resides, species i has a success rate of displacing species j of η_{i, j} (η_{i, j} + η_{j, i} = 1), which is negatively correlated with the differences in colonization rates—i.e., better colonizers (measured by c_i) are worse competitors (measured by η_{i, j}). The dynamics of species i’s site occupancy are given by

$$ \frac{d{p}_i}{dt}={c}_i{p}_i\left(1-{p}_i-{p}_j-{m}_i+{p}_j{\eta}_{i,j}-\frac{c_j}{c_i}{p}_j{\eta}_{j,i}\right) $$

(34)

$$ {\eta}_{i,j}=\frac{\omega_i}{\omega_i+{\omega}_j} $$

(35)

$$ {\omega}_i=\exp \left(-\beta {c}_i\right). $$

(36)

We do not consider preemption competition, as do Calcagno et al. (2006), and thus, we also do not generally find limits to dissimilarity in our model, as they do.

As in model 2 above, the condition for stable coexistence of the two species in this model is that each species must have positive growth in occupancy (dp_i/dt > 0 for species i) when it is rare (when p_i ≈ 0 for species i) and the other species is at the equilibrium it would reach in the absence of the other species, denoted p_i^** for species i. From Eq. (34), single-species equilibrium for species i is (Levins 1969)

$$ {p_i}^{\ast \ast }=1-{m}_i. $$

(37)

Thus, coexistence requires

$$ 0<1-{m}_i-{p_j}^{\ast \ast }{\eta}_{j,i}\left(1+\frac{c_j}{c_i}\right). $$

(38)

Equation (38), for both species, can be re-written as

$$ {\eta}_{j,i}\left(1+\frac{c_j}{c_i}\right)<\frac{1-{m}_i}{1-{m}_j}<\frac{1}{\left(1+\frac{c_i}{c_j}\right){\eta}_{i,j}}. $$

(39)

This condition is analogous to Eqs. (3a), (19), and (29) from the other models, under most parameter values. As the two species become similar in niche (i.e., c_i approaches c_j, implying that η_{j, i} and η_{i, j} both approach 0.5, and ${\eta}_{j,i}\left(1+\frac{c_j}{c_i}\right)$ approaches $\frac{1}{\left(1+\frac{c_i}{c_j}\right){\eta}_{i,j}}$), the range of relative mortality rates ($\frac{1-{m}_i}{1-{m}_j}$) allowing coexistence becomes smaller (Fig. 3h).

The exception occurs when either c_i or β is relatively small (e.g., c_i < 0.05 or β < 20). In such cases, the competitive disadvantage of species i’s lesser colonization ability becomes harder, rather than easier, to offset via better competition (higher η_{i, j}) as the difference in niche ($\frac{c_j}{c_i}$) magnifies. The reason for this is that lower values of β carry less of a competition advantage per unit decrease in colonization ability. However, assuming that the species coexist with no stress ensures that β has a relatively high lower-bound $\beta >\frac{\log \left({c}_j\right)-\log \left({c}_i\right)}{c_j-{c}_i}$ [by Eq. (39)]. For instance, if c₁ = 0.2 and c₂ = 0.1, β > 7. Thus, it is a relatively limited range of intermediate β values under which Eq. (39) does not hold. Note also that the classic competition-colonization model (e.g., Tilman 1994), in which the poorer colonizer always wins in local competition, would be equivalent to β = ∞ (i.e., η_{i, j} = 1, η_{j, i} = 0, when species i is the inferior colonizer, and they make the additional assumption that m_i0c_i = m_j0c_j).

Assuming parameters are such that the range of relative mortality rates decreases as the species become similar in niche (as in the other models), it is straightforward to show that the stressor intensity causing competitive exclusion of one of the species has an upper bound, which approaches zero as the niches approach identity (i.e., as c_i approaches c_j). For instance, suppose the stressor intensity (E) has a linear effect on the mortality of each species

$$ {m}_i={m}_{i0}+{\gamma}_iE $$

(40)

where m_i0 is the baseline natural mortality for species i. Coexistence Eq. (39) becomes

$$ {\eta}_{j,i}\left(1+\frac{c_j}{c_i}\right)<\frac{1-{m}_{i0}-{\gamma}_iE}{1-{m}_{j0}-{\gamma}_jE}<\frac{1}{\left(1+\frac{c_i}{c_j}\right){\eta}_{i,j}}. $$

(41)

If ${\gamma}_i=\left(\frac{1-{m}_{i0}}{1-{m}_{j0}}\right){\gamma}_j$, $\frac{1-{m}_{i0}-{\gamma}_iE}{1-{m}_{j0}-{\gamma}_jE}$ does not change as E increases—this is the equal sensitivity scenario. If ${\gamma}_i>\left(\frac{1-{m}_{i0}}{1-{m}_{j0}}\right){\gamma}_j$, stress competitively disadvantages species i and the intensity (E_c) needed to competitively exclude species i is bounded by

$$ 0<{E}_{\mathrm{c}}\le \frac{\left(1-{m}_{i0}\right)\left(1-{\eta}_{j,i}{\eta}_{i,j}\left(2+\frac{c_j}{c_i}+\frac{c_i}{c_j}\right)\right)}{\left({\gamma}_i-{\eta}_{j,i}\left(1+\frac{c_j}{c_i}\right){\gamma}_j\right)}. $$

(42)

These bounds are analogous to Eqs. (4), (21a), (21b), and (33) in the other three models. As c_i approaches c_j, η_{i, j} and η_{j, i} approach 0.5. Thus, $1-{\eta}_{j,i}{\eta}_{i,j}\left(2+\frac{c_j}{c_i}+\frac{c_i}{c_j}\right)$ approaches zero, as does the upper bound on E_c.

Table 1 Model parameters