A Stable Ecosystem Is Both

PLoS One. 2020; 15(iv): e0228692.

Multifariousness increases the stability of ecosystems

Francesca Arese Lucini, Information curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – review & editing,¹ Flaviano Morone, Conceptualization, Data curation, Formal analysis, Investigation, Software, Supervision, Writing – review & editing,ⁱ Maria Silvina Tomassone, Conceptualization, Formal analysis, Methodology, Resources, Software, Supervision, Validation, Writing – review & editing,^two and Hernán A. Makse, Conceptualization, Information curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original typhoon, Writing – review & editing ^1, ^*

Francesca Arese Lucini

¹ Levich Institute and Physics Department, Urban center College of New York, New York, NY, Usa

Flaviano Morone

¹ Levich Institute and Physics Department, City Higher of New York, New York, NY, U.s. of America

Maria Silvina Tomassone

² Department of Chemic and Biochemical Applied science, Rutgers University, Piscataway, NJ, Us of America

Hernán A. Makse

^one Levich Constitute and Physics Department, City College of New York, New York, NY, U.s.

Judi Hewitt, Editor

Received 2020 Jan 15; Accepted 2020 Feb 27.

Abstract

In 1972, Robert May showed that variety is detrimental to an ecosystem since, as the number of species increases, the ecosystem is less stable. This is the and so-called diversity-stability paradox, which has been derived by considering a mathematical model with linear interactions between the species. Despite being in contradiction with empirical testify, the diversity-stability paradox has survived the test of fourth dimension for over xl+ years. In this paper nosotros first prove that this paradox is a conclusion driven solely past the linearity of the model employed in its derivation which allows for the neglection of the fixed point solution in the stability analysis. The linear model leads to an ill-posed solution and along with it, its paradoxical stability predictions. We so consider a model ecosystem with nonlinear interactions between species, which leads to a stable ecosystem when the number of species is increased. The saturating not linear term in the species interaction is analogous to a Hill function appearing in systems like gene regulation, neurons, diffusion of information and ecosystems The exact stock-still bespeak solution of this model is based on grand-core percolation and shows that the paradox disappears. This theoretical result, which is exact and non-perturbative, shows that diverseness is benign to the ecosystem in agreement with analyzed experimental prove.

Introduction

The relationship between species diversity and stability in ecosystems has been extensively studied in the literature [1–sixteen]. The pioneering study led by Sir Robert May [ane] predicted that the more diverse an ecosystem is, the more than unstable information technology is. May'south claim resonated powerfully amid ecologists as it contradicted the biological principle that bang-up variety of species (and genes) promote ecosystem stability in the face of external stress, and this foundation turned May's merits into a paradox, referred to every bit the variety-stability paradox. For well-nigh forty years this paradox was not able to be refuted, despite prove showing that ecosystems in nature that accept a high degree of diversity tend to be more than stable [14]. It was only until recently that the concept of loftier diversity linked to stability started to emerge; supporting the idea that increasing species diversity is positively correlated with increasing stability at the ecosystem-level [13, xv, 17, xviii] and negatively correlated with species-level stability due to declining population sizes of individual species [14]. However, so far there has not been a theoretical proof that demonstrates mathematically the reason why this occurs. In this article, we show that, using a nonlinear interactions model, the arrangement becomes more stable when there is more species variety, a statement that differs from the results of the linear model derived from May.

In section 2 we get-go derive the diversity-stability paradox explicitly past solving the linear model studied by [9–13], which follows the aforementioned reasoning as [i]. Nosotros show that the solution of the linear model diverges for sure values of the interaction species and thus, it'southward sick posed. In Section III we advise a nonlinear solution based on a model proposed past [19–22] and developed in [16] by analyzing the solution of this model, we illustrate that when the interaction strength between mutualistic species is positive and strong, more than species in the ecosystem survive. Both the solutions of the linear and nonlinear model are applied to real world ecosystems with positive mutualistic interaction terms between species, then to requite a practical example of the two unlike weather condition for stability. In Section 4 we present a give-and-take of the results. We volition see that the experimental evidence will back up the use of the nonlinear model as a more than accurate description of the ecosystems' stability.

Solution of the linear model for arbitrary adjacency matrix

We volition first show the solution of the linear model diverges for given values of the interaction species.

In general the evolution of species abundances ten _i(t) in a ecosystem can be described by dynamical equations of the grade:

$\begin{matrix} \dot{x_{i}} (t) = f_{i} (x_{i}) + g_{i} (x_{ane}, x_{2}, . . . . ., 10_{n}) . \end{matrix}$

(1)

The linear model for ecological networks is described by an adjacency matrix A _ij (with A _ij = 1 if i and j are connected by a network link, and A _ij = 0 otherwise), and linear interactions betwixt species given by:

$\begin{matrix} g_{i} (x_{1}, . . . ., {ten}_{n}) = \sum_{j = 1}^{n} γ_{i j} A_{i j} x_{i} x_{j} . \end{matrix}$

(2)

The dynamics of species densities x _i is and so described past the following dynamical system of equations:

$\begin{matrix} \dot{10_{i}} (t) = b_{i} x_{i} - due south {ten}_{i}^{ii} + \sum_{j = i}^{N} γ_{i j} A_{i j} x_{i} 10_{j}, i \in {1, \dots, Due north} . \end{matrix}$

(3)

where b _i > 0 is the growth rate of species i, s is the cocky limitation term representing the self-interaction of species, that we set equal for all species, γ _ij is the strength of the interaction between species i and j, and Due north is the total number of interacting species. The fixed bespeak equations $\frac{d x_{i} (t)}{d t} = 0$ admit a footling solution x* = 0, which represent the extinction of all species, and a not-trivial solution ten* ≠ 0 which, in implicit form, is given by the following linear organisation:

where Γ_ij = γ _ij A _ij. If $s \notin Spec (\hat{Γ})$ , then the matrix (sI − Γ) is invertible and nosotros can write the solution x* as:

where μ _a are the eigenvalues of $\hat{Γ}$ and 50 ^a, R ^a the corresponding left and correct eigenvectors.

Eq (5) shows that the fixed point solution of the linear model has a singularity whenever there is an eigenvalue of Γμ _a = southward, for some a ∈ {1, …, N}. In particular, we can think of a situation where the ecosystem is going through a period where the forcefulness of the interactions γ _ij is increasing. In this instance the linear model becomes sick-defined when the largest eigenvalue of the matrix $\hat{Γ}$ equals s, μ _max = south, considering the species densities diverge, ${ten}_{i}^{*} \to \infty$ (encounter Fig 1).

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The solution of the linear model of Refs. [9, 10] shows the dependence of the average density of species 〈x*〉 as a role of the ratio s/γ, as given past Eq (5).

For modest values of the interaction forcefulness γ, the arrangement is in the feasible and stable fixed point 〈ten*〉 > 0 (lower branch of the full line in the pinnacle correct quadrant). Increasing γ, at fixed s, the species density 〈ten*〉 increases post-obit the total cherry-red line, and somewhen diverges at the critical point γ _c predicted past the linear model to be γ _c = s/μ _max. For south/γ < μ _max, the nontrivial stock-still bespeak is negative, 〈ten*〉 < 0, and unstable (dashed line), so that the only stable fixed betoken is the collapsed state 〈x*〉 = 0 (red dot). Thus, the linear model of mutualism predicts the collapse of the ecosystem equally the instantaneous extinction of an infinite number of mutualists at the diverging point s/γ _c.

Through the assay of the solution of the fixed point we can also notice that the condition of stability; Eq (5) is feasible if and only if the densities $x_{i}^{*}$ are all positive. This is certainly true when μ _max is smaller than south. Only when it is close to information technology (i.due east. 0 < southward − μ _max ⪡ 1), the sum on the r.h.s of Eq (5) is dominated by the term containing μ _max, thus giving

$\begin{matrix} x^{*} \approx \frac{L^{(max)} \cdot b}{s - μ_{max}} R^{(max)} for μ_{max} \to s . \end{matrix}$

(6)

Since $\hat{Γ}$ is an irreducible matrix with non-negative entries, then, by the Perron-Fronebius theorem, the right and left eigenvectors R ^(max) and 50 ^(max) have all positive components, so the vector x* does accept strictly positive components, too. On the contrary, when μ _max becomes larger than south, all densities plough negative, $x_{i}^{*} < 0$ , and the solution x* becomes unfeasible hence the stability status is given by s − μ > 0. Next we will evidence that this condition for stability tin besides be found via a local stability analysis of the dynamical system.

Local stability of the fixed betoken solution

The criterion for ecosystem stability is given by the sign of the largest eigenvalue of the stability matrix calculated at the fixed signal x* for the dynamical organization of Eq (3), which is expressed by the Jacobian $J_{i j} (x^{*}) = \frac{\partial {\dot{10}}_{i}}{\partial x_{j}} |_{x^{*}}$ . Negative eigenvalues indicate that the system is stable. That is, if one of the eigenvalues of the Jacobian is positive, the average may be positive, or zero, and in that case, the arrangement is non stable.

The Jacobian is expressed by:

$\begin{matrix} J_{i j} (x^{*}) = \frac{\partial {\dot{10}}_{i}}{\partial x_{j}} |_{{ten}^{*}} = - x_{i}^{*} (s δ_{i j} - Γ_{i j}) . \end{matrix}$

(7)

The eigenvalues of J _ij(x*) are not only related to those of Γ_ij, due to the multiplicative term ${ten}_{i}^{*}$ in Eq (7). However, when $\hat{Γ}$ is symmetric, we can utilize the following strategy to infer the crucial properties of the eigenvalues of $\hat{J}$ . Offset, we ascertain the matrix $X_{i j} = 10_{i}^{*} δ_{i j}$ , and nosotros set M _ij = −s(δ _ij − Γ_ij), so that $\hat{J} = \hat{X} \hat{M}$ . Adjacent, we observe that $\hat{J}$ is similar to the symmetric matrix $\hat{\tilde{J}} = {\hat{Ten}}^{1 / 2} \hat{M} {\hat{X}}^{1 / 2}$ , so $\hat{J}$ and $\hat{\tilde{J}}$ have the same eigenvalues. The crucial point is that $\hat{\tilde{J}}$ and $\hat{M}$ are congruent matrices, and therefore, past Sylvester's constabulary of inertia, they take the aforementioned number of positive, negative, and zero eigenvalues. Therefore, if $\hat{M}$ has all negative eigenvalues, $\hat{\tilde{J}}$ too has all negative eigenvalues, hence, by similarity, too J has all negative eigenvalues. On the other hand, when μ _max = s, we know $\hat{M}$ has a zero eigenvalue, but then also the Jacobian $\hat{J}$ must accept a zero eigenvalue, which means that the solution 10* is not stable anymore, as we anticipated in the previous section. It is interesting to observe for which cases of the ratio $\frac{s}{γ}$ the organization is not stable. Fig 2 shows a plot of the sign of the maximum eigenvalue as a function of the interaction term taken for the real networks numbered 1 to 9 in Table i.

Table 1

Details of the 9 mutualistic networks used in the stage diagram of Figs iii and 4.

Net #	Network type	Plants	Animals	Breadth	Location	Ref.
one	Plant-Seed Disperser	31	9	Tropical	Papua New Guinea	[23]
ii	Constitute-Pollinator	91	679	Temperate	Japan	[24]
three	Plant-Pollinator	42	91	Temperate	Australia	[25]
four	Institute-Pollinator	23	118	Artic	Sweden	[26]
5	Plant-Pollinator	11	eighteen	Artic	Canada	[27]
6	Constitute-Pollinator	14	thirteen	Temperate	Mauritius Island	[28]
7	Plant-Pollinator	7	32	Temperate	USA	[29]
viii	Institute-Pollinator	29	86	Artic	Canada	[30]
9	Plant-Seed Disperser	12	14	Temperate	U.k.	[31]

An external file that holds a picture, illustration, etc. Object name is pone.0228692.g002.jpg

Plot of the sign of the maximum eigenvalue $λ_{chiliad a x}^{M}$ of $\hat{M}$ every bit a function of the interaction for real networks and constant value of the self limitation term s.

The inset of the figure indicates the number of the real network (1-9) shown in Table 1. The sign of the maximum eigenvalue $λ_{k a 10}^{M}$ of $\hat{M}$ changes as a part of $\frac{due south}{γ}$ where γ is the coupling term and this change of sign occurs at μ _max = s where μ _max is the maximum eigenvalue of the matrix Γ of the corresponding network. This is represented past the dotted line in this figure, therefore the value of γ for which $R east (λ_{m a x}^{G}) = 0$ coincides with the condition of the singularity obtained with the solution to the fixed point equation discussed in Section, i.e. γ for which Re(μ _max) = 0 where $Γ = γ \hat{A}$ , for $\hat{A}$ existence the adjacency matrix. The networks analyzed are labeled according to the references in Table i. (Observe that networks iv and 8 are overlapping).

To explicitly re-derive the validity of the solution of the fixed point equation we analyzed the spectrum of the matrix $\hat{M}$ instead of the spectrum of the Jacobian directly in guild to avoid incurring into ciphering issues at the singularity of x*. The condition to be verified is the sign of the real office of the maximum eigenvalue of $\hat{J}$ . If this sign is negative the system is stable. If nothing or positive, the organization is unstable. In Fig 2 nosotros fix the cooperation value to the average of all interactions of the organisation γ _ij = γ and plot the sign of the maximum eigenvalue of matrix $\hat{M}$ ( $λ_{m a ten}^{M}$ ) for 9 unlike ecosystems shown in Table ane as a role of $\frac{s}{γ} .$ The figure shows that $λ_{yard a 10}^{M}$ can be positive, negative or null when γ is varied, which in this instance is a scalar, hence, according to what we previously noted, $\hat{J}$ will also have a zero maximum eigenvalue which occurs at the critical status when μ _max = s and changes sign for varying γ. According to Eq 7, the maximum eigenvalue of $\hat{J}$ ( $λ_{1000 a 10}^{J}$ ) volition have the aforementioned sign of $λ_{m a x}^{Yard}$ , hence the condition for stability is given by $λ_{m a 10}^{J} < \frac{southward}{γ}$ . Annotation that $λ_{thousand a x}^{J} \neq μ_{1000 a ten}$ where μ _max is the maximum eigenvalue of Γ but the ii conditions of stability are equivalent since the point in which the sign of the eigenvalues of $\hat{J}$ and $\hat{Γ}$ alter are equivalent, as shown in Fig 2.

It is worth mentioning that, as shown in Fig two, that $λ_{m a ten}^{J}$ changes sign as a function of the interaction term γ. The point at which the existent office of the eigenvalue $λ_{grand a x}^{J}$ becomes negative indicates the position of the transition from the stable stage (x > 0), to the unstable stage (x < 0) (as shown in Fig 1). In particular, nosotros note that the status for stability inferred from the analysis of the jacobian coincides exactly with the disquisitional point obtained directly from the analysis of the required positivity of the density of the fixed signal x* brought out in the previous section.

Status of stability through Wigner's police force

May'due south approach, which is usually adopted as well from more recent studies of linear model [9, 10] is the application of the analog of Wigner'due south semicircle law for asymmetric matrices, the circular police force [1]. This police force states that for self regulating systems where the diagonal elements are such that J _ii = s < 0, and the off-diagonal elements J _ij are independent and identically distributed random variables, with zero hateful and variance σ ⁱⁱ, the eigenvalues of $\hat{J}$ lie in a disk of radius $r_{J} = σ \sqrt{North}$ for Due north → ∞ centered in −s.

Ecological systems are usually only sparsely connected. Hence, both May [1] and [9, ten] introduce the connectance C in their calculations; C measures the probability that species interact, consequently the probability of no interaction is given by i − C. In this case the circular law states that $r_{J} = σ \sqrt{N C}$ . Applying the condition for stability that $R e (λ_{chiliad a ten}^{J} < 0)$ gives:

$\begin{matrix} \begin{matrix} R e (λ_{m a ten}^{J}) \approx \sqrt{Due north C} σ - south < 0 \\ \sqrt{N C} < \frac{s}{σ} . \end{matrix} \end{matrix}$

(8)

For N → ∞, the radius of the disk and hence the maximum eigenvalue is $λ_{g a x}^{J} \approx \sqrt{Northward C}$ and σ can be seen as the average interaction strength γ hence $λ_{max}^{J} < s / γ$ is the condition for the local stability of the feasible equilibria x* ≠ 0. In other words, when $λ_{max}^{J} > s / γ$ , the nontrivial fixed bespeak x* ≠ 0 is unstable (and unfeasible since the average species density 〈10*〉 would be negative: 〈ten*〉 < 0). This stability of the average 〈x*〉 < 0 is shown in Fig 1. These results atomic number 82 to the paradox, since when N increases the system becomes more unstable.

Thus, in the stable feasible region of the linear model Eq (3), the status:

$\begin{matrix} λ_{max}^{J} < s / γ (condition of stability in the linear model), \end{matrix}$

(ix)

holds true.

We have and so shown that all three methods of report of stability for the linear model produce the same stability status: the so called diversity-stability paradox.

We test the stability condition by analyzing 9 existent mutualistic networks (with positive interactions) compiled from available online resources and detailed in Table i.

Nosotros are able to test the stability phase diagram since for these networks the parameters of the model are provided, in particular the strength of interactions which is the parameter that control the stability of the linear ecosystem via Eq (nine). Fig 3 shows the phase diagram for stability predicted by Eq (9) in terms of the values of $(southward / γ, λ_{max}^{A})$ for each of the 9 analyzed mutualistic ecosystems.

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Phase diagram of the linear model equation which plots the largest eigenvalue of the adjacency matrix μ _max versus the ratio southward/γ for the 9 empirical mutualistic networks explained in Table 1.

All the networks lie in the unstable region μ _max > s/γ, and hence they do non satisfy the condition of the linear model μ _max < s/γ, which is necessary to accept a viable (i.e. 〈x*〉 > 0) and stable solution. Hence the linear model of Eq (3) predicts that these nine existing ecosystems should indeed collapse (i.e. 〈10*〉 = 0 for all of them) in contradiction to the fact that they are existent viable ecosystems present in nature.

We notice that the stability condition of the linear model, Eq (nine), is non satisfied past these real ecosystems. That is, all real mutualistic networks are located in the unstable region $λ_{max}^{A} > s / γ$ , as seen in Fig 3, and thus, according to the linear model (3), all systems should collapse. Below we will explicate in more than item the nonlinear dynamical model, which predicts opposite results for the condition of stability with respect to the linear model and explains the existence of the ix real mutualistic networks, suggesting the nonlinear model as a more adequate report of ecological systems.

Stability for the nonlinear functional response

Most of the studies on stability for ecosystems have been done using the linear model explained in Section, mainly because one can discover an analytical solution to the fixed point equation and the stability condition is directly related to the eigenvalues of the adjacency or jacobian matrix. On the other hand, our grouping has previously analyzed mutualistic ecosystems using network theory and found an verbal solution of the nonlinear Type 2 functional responses ([16]), where the ratio of species consumed as a function of the species' population is expressed by a term that saturates featuring a more realistic situation when, even if the size of the species is increased, the number of species depleted remains abiding at saturation. This behavior is common for the description of cistron regulation, neurons, diffusion of information and ecosystems as presented in their article. The dynamics of species densities, 10 _i(t), interacting via the network A _ij, is described past the following set of nonlinear differential equations [19–22, 32]:

$\begin{matrix} \begin{matrix} \dot{x_{i}} (t) = - x_{i} d - s x_{i}^{two} + γ \sum_{j = one}^{N} A_{i j} \frac{{ten}_{i} x_{j}}{α + \sum_{j = 1}^{N} A_{i j} x_{j}}, \\ i \in {i, \dots, N}, \end{matrix} \end{matrix}$

(10)

where d > 0 is the decease charge per unit of the species, s > 0 is a cocky limitation parameter directed to the limitation of species' growth when x _i exceeds a certain value, α is the one-half saturation constant, and γ > 0 is the average interaction strength of the nonlinear interaction term. All these dynamical parameters (γ, d, s, α) are discussed in depth in previous works [nineteen–22, 32].

To study the stability of the solution, one has to first discover the nontrivial fixed point 10* ≠ 0, which has been obtained in [16]. Using a simple logic approximation on the saturating term the solution of the dynamical equations for constant interaction term γ is given by:

$\begin{matrix} \begin{matrix} y_{i}^{*} & = \sum_{j = 1}^{N} A_{i j} Θ (y_{j}^{*} - {Yard}_{γ}), \\ K_{γ} & = \frac{α s (γ + d)}{{(γ - d)}^{2}}, \end{matrix} \end{matrix}$

(11)

where $y_{i}^{*}$ represents the reduced density, the Heaviside function Θ(y) = ane if y > 0 and nothing otherwise, and 1000 _γ is the threshold on the mutualistic do good, where the subscript emphasizes its main dependence on γ, the interaction term. The interaction term can exist rewritten as a Loma part of degree n = 1. For degree due north → ∞, the Loma part can be replaced past a Heaviside function. Even if one approximates the interaction term to the Heaviside function, information technology is possible to compare the solution given by verbal numerical simulations to the solution of the approximated method of Eq (11) and testify consistency within a 12.5% error.

We have said that Yard _γ in Eq (11) represents the threshold of the Θ-function; species i interact with species j just if the reduced density $y_{j}^{*}$ is above G _γ.

When γ is modest, which ways that the interaction is weak, One thousand _γ is large and a smaller number of species j survive (east.g a pocket-size number of species have densities such that $x_{j}^{*} > K_{γ}$ for weak γ). For γ = γ _c, $M_{γ_{c}}$ is too big so that no mutualistic do good between species can be exchanged; at this point the system collapses to the fixed point solution ten* = 0. At this stage the $1000_{γ_{c}}$ is given by

In contrast, when the interaction is potent the threshold for the mutualistic benefit G _γ is low and a large number of interacting species j survive. Afterward a trimming procedure of the species in the network, the solution of Eq (xi) is shown to be:

$\begin{matrix} \begin{matrix} y_{i}^{*} = number of links of species i \\ to species in the {One thousand}_{γ} - core \equiv {Due north}_{i} (K_{γ}) \end{matrix} \end{matrix}$

(13)

where G _γ-core of a network is the subset of nodes in a network that have degree of at to the lowest degree Thousand _γ (integer number), therefore its the most continued subgroup of the graph. Co-ordinate to Eq (13), the tipping point of a mutualistic ecosystem, whose motion is described by Eq (ten), is given by the extinction of the species that belong to the $K_{γ_{c}}$ -core of a network, expressed in Eq (12) which allows a relation between the dynamical properties of the mutualistic network and a topological invariant of the system, the k-core. The density $y_{i}^{*}$ is positive and therefore ${Northward}_{i} (G_{γ})$ is besides positive. This condition tin only be satisfied if $K_{γ} < k_{cadre}^{max}$ , where ${yard}_{core}^{max}$ is the maximum k-cadre of the network. Consequently, if this status is non satisfied the organization collapses to the fixed point solution ${ten}_{i}^{*} = 0$ , and the tipping point, described in Eq (12), occurs when

which relates the dynamical parameters of (10) to the structural properties of the networks. As a consequence, the stability status is given by:

$\begin{matrix} \begin{matrix} k_{core}^{max} > K_{γ} \propto s / γ (condition of stability in the nonlinear model), \end{matrix} \end{matrix}$

(fifteen)

According to the solution of the nonlinear model, the larger the k-core number $k_{core}^{max}$ (i.eastward., the more one thousand-shells in the network) the larger the resilience of the arrangement against external global shocks that reduce the interaction strength γ. With the solution Eqs (eleven) and (13) of the fixed point equations, nosotros tin now study the local stability of the type Two dynamic equations by analyzing the Jacobian of the stability matrix

$\begin{matrix} J_{i j} ({ten}^{*}) = \frac{\partial {\dot{x}}_{i}}{\partial {ten}_{j}} |_{ten = x^{*}} . \end{matrix}$

(sixteen)

To guarantee the stability of the fixed point one has to verify that the real office of the eigenvalues of (7) are all negative. The eigenvalues $λ_{i}^{J}$ of J are:

$\begin{matrix} λ_{i}^{J} = - γ \frac{N_{i} (G_{γ})}{M_{γ} + N_{i} ({Yard}_{γ})}, i = ane, \dots, N, \end{matrix}$

(17)

and we can easily see that all eigenvalues are negative. The maximum eigenvalue, which is obtained when the nodes (i.east species) of the network have fewest number of edges with the K _γ-core, is given by $λ_{m a 10}^{J} = - γ {(K_{γ} + 1)}^{- i}$ , is evidently ever negative; therefore when the solution is viable, according to Eq (15), it is always stable.

Fig 4 plots the phase diagram of ecosystem stability in the space $({Chiliad}_{γ}, {yard}_{core}^{max})$ predicted by the nonlinear model and features the 'tipping line', which separates the feasible stable stage of the nonlinear model (15) from the collapsed stage $k_{core}^{max} < {Grand}_{γ}$ . Here we plot the values of $(K_{γ}, k_{core}^{max})$ obtained from the 9 existent mutualistic networks illustrated in Table 1, for which these two parameters have been measured in the literature [9, 21]. All mutualistic networks lie in the stable feasible region situated below the tipping line, in agreement with the dynamical theory of the nonlinear model, and in dissimilarity to the result of the linear model. We take and so shown that by taking into account the actual fixed point solution in the stability analysis, forth with the saturation event of the interaction term, this resolves the diversity-stability paradox.

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Plot of the stage diagram of the solutions of the nonlinear model.

The stability diagram equally a role of K _γ is computed using the exact solution (11) of the nonlinear dynamical organization for the same 9 empirical mutualistic networks used in Fig 3. The tipping line is plotted co-ordinate to the nonlinear model which predicts that the tipping bespeak is given for ${chiliad}_{core}^{max} = {Thou}_{γ}$ . All the networks lie in the region ${thousand}_{core}^{max} > K_{γ}$ and thus they are stable, equally they should.

Discussion

We accept presented two different approaches for the study of the stability status in ecosystems and have seen that in the case of a fully connected network, the linear approach [1, 9, x, 33] leads to counterintuitive predictions which are in contrast with the exact solution of the full nonlinear model (Eq (10)).

The linear model of Eq (3) contains the diversity-stability paradox [1], for which, a more diverse ecosystem is closer to the point of turning unstable [ane]. According to the linear model, increasing the diversity has, in general, a destabilizing effect on the mutualistic ecosystem, since it requires the interacting term γ to exist smaller. We can and so state that the linear stability assay of the ecosystem features two main features: first, it cannot detect the tipping point of the arrangement collapse Eq (11) and second, the stability analysis [1, 9, 10, 33] is in contrast with the exact solution of the full nonlinear model (Eq (10)) described by a saturating function. The prove for such controversy is provided in Fig 3, which plots the maximum eigenvalues of ix real networks which are known to be stable as lying in the unstable regime. Co-ordinate to the linear model then, all real mutualistic networks of Table 1 would be collapsed, and, for the 9 networks studied, this is not the instance.

On the other hand, the result of considering a saturation Hill function in the interaction term, (Eq x) leads to the reverse stability condition: in a fully connected mutualistic network, Eq (15) read N > s/γ. In this case, by increasing the species multifariousness N, the condition N > due south/γ is easier to satisfy. Similarly, by increasing the mutualistic interaction γ, the stability status North > s/γ is also easier to satisfy. In decision, the nonlinear model predicts both diversity and mutualism to have a stabilizing effect on the whole ecosystem and correctly predicts that the analyzed ecosystems should be feasible as shown in Fig iv.

Thus, the effect of the nonlinear model is then crucial to predict the stability and feasibility of the ecosystem.

Furthermore, the linear approximation predicts that more diverse systems (i.east. systems with larger λ _max due to either larger connectivity k or larger number of species), are closer to collapse. Analytically, the origin of May's paradox can be traced dorsum to a mathematical singularity in the linear model at the tipping point (Fig 1): the density $x_{i}^{*}$ diverges at s/γ _c = μ _max, and and then collapses instantaneously to the country $x_{i}^{*} = 0$ . This singularity is absent in the nonlinear model due to the saturation upshot of the nonlinear interaction term, thus resolving the paradox of the linear model for 2 main reasons. First, because Eq (fifteen) predicts that the larger the mutualistic strength γ, the more than stable the system is. Second, increasing diversity by the number of connections or the number of species, increases the maximum m-core, (or at least leaves information technology unchanged), thus increasing the robustness of the system. Therefore, stronger mutualistic interactions and augmented multifariousness stabilize the organization, as confirmed by existent ecosystems in Fig 3. Thus, all these reasonings indicate the importance of considering the full set of nonlinear interactions when reaching conclusions on the stability of ecosystems. For instance, recent studies [10] take used the linear model to analyze the stability of the microbiota, and have concluded that mutualism in bacteria species is detrimental to the ecosystem. Such a conclusion would be reversed if one were to employ the nonlinear model to analyze the information.

Funding Statement

Research was sponsored by NSF-IIS 1515022 and NIH-NIBIB R01EB022720.

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