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The impact of life-history strategies on the stability of competitive ecological network

In natural ecosystems, species often compete with multiple species to survive and acquire resources. These species can be characterized by their different growth profile and life history strategies, namely r and K selected species (Williams et.al., The journal of wildlife management, 2013). In this paper, we examine the role of different life-history strategies on the stability of the competitive community along with network complexity. We obtain that the r-selected species enhance community stability and resilience; on the other hand, K-selected species show the opposite trend. Network connectance shows a consistent trend of increasing stability; on the other side, the number of species shows a context-dependent effect on stability. Stability decreases with species number when the proportion of r-selected species is low or moderate and shows precisely the opposite trend when the ratio of r-selected species is high. We also show that our result is robust irrespective of network structure.


Introduction
The intrinsic mechanisms behind the stabilization of an ecological community is a common query of ecologists since the pioneering work of May (May, 1972) around fifty years ago. He revealed that complexity should be a regulating factor for the stability of an ecological system. Henceforth, the complexity and stability of ecological communities is a long-standing unsettled issue and still inconclusive. Various debates and cross-debates exist (McCann, 2000;Elton, 2000;Nunney, compete for pinyon pine seeds (Christensen and Whitham, 1993). Also, birds and insects compete for nectar (Carpenter, 1979).
It has been paid very little attention to the role of non-linear density dependence on intraspecific growth rate and its impact on the complexity-stability relationship. Several species are connected through various ecological relationships in ecological communities, forming a complex network of inter-specific interactions. Depending on interaction types, there are multiple categories of communities (Food web, Mutualistic Community, Host-Parasite network, Competitive community) that influences the dynamics and structure of a community (Kawatsu and Kondoh, 2018;Mougi and Kondoh, 2012;Allesina and Tang, 2012).
In general, most of the r-selected species actually lives in harsh environments such as polar region where the unpredictability of the environment and the seasonal variation in food abundances is very high. So most species found in the polar region are r-strategists, as example, red marine shrimps and penguins. In these regions, the change in biodiversity is low. (Krebs and Davies, 2009;Kozlowski, 2012) On the other hand, environmental fluctuation is very low in the tropical regions and most of the species found in these regions are k-selected which are highly competitive. As a result, the small change in tropical habitat causes extinction of many of the species and change in biodiversity is very high.
There are several kinds of stability measures in ecological studies, such as local stability, resilience, persistence, and so on (Landi et al., 2018). Local stability measures the system's tendency to return to a feasible equilibrium (where all species co-exists) after a small perturbation. Resilience describes how fast the state of a locally asymptotically stable system returns to its equilibrium following a perturbation away from the equilibrium. We used 'local stability' and 'resilience' to measure stability for random, scale-free, and small-world network structures. In the random network structure, any link among N species occurs with the same probability (Newman, 2018).
In the case of scale-free networks, most nodes have one or a few links, but a few nodes are extremely well connected (Barabási and Albert, 1999) and small-world networks typically show high clustering coefficients and low characteristic path length (Watts and Strogatz, 1998). (Clustering coefficient defines how well connected the neighbors of a considering node and characteristic path length represent the distance between any two randomly selected nodes (Newman, 2018)).
Our goal is to investigate the effect of r and K-selection strategies on the stability of a competitive community in terms of their density-dependent growth rate profile. In this study, we derived a theoretical condition for stability in the case of random network structure with relaxed assumptions.
We tested the generality of the condition derived by performing numerical simulations. It is hard for scale-free and small-world type networks to derive theoretical conditions, so we only perform numerical simulations. To do this, we generalized the Lotka-Volterra model by considering nonlinearity in species' relative growth rate profile. We hypothesize that r and k-selection strategies may have a remarkable impact in maintaining stability in a competitive community due to their unlike type of growth profile.
The paper is organized as follows. The section 2 contains the mathematical and biological concept behind the formulation of our model. Then we analyze the local stability through the random matrix theory. In section 3, we simulate our result through numerical technique, and in section 4, we discuss some important results and ecological aspect of our result on Finally, we end with a conclusion in section 5.

Mathematical Model
Theoretical ecologists have plenty of choices to select a mathematical model which shows functional concavity. This is a known fact that in most of the r-strategist species, the RGR-density relationship is concave in nature. But in comparison, K-strategy species exhibit convex trajectory for RGRdensity relationship. Sibly et al. Concluded that most of the RGR density relationship are concave upwards in nature. The theta-logistic model has certain limitations in predicting the reliable estimate of the instantaneous growth rate (r) and density regulation parameter (θ), while the uncertainty present in the system and carrying capacity is time-dependent (Clark et al., 2010).
But at the same time, the theta-logistic model may be a good choice to understand the mechanism of the proportion of r and K-selected species.
Life-history demographic parameters play an influential role in the specific growth rate, and in connection to this, some studies attempted to investigate the nonlinear density-dependent pattern (Saether and Engen, 2002;Fowler, 1981). These works reveal that, in the case of r-selected species, greater r max (maximum intrinsic growth rate) values are expected to be observed compared with the K-selected species, and the reduction rate is high at low densities. A high intrinsic growth rate makes the r-selected growth faster toward carrying capacity. At the same time, K-selected species grow relatively slow with weak density dependence at small population sizes. These can be well explained through the difference in several morphological, behavioral, demographic character-istics. The r-selected species are generally small-sized, lives in favorable breeding habitats, have early maturity, high fecundity, low survival rate, and high reproductive capacity. However, the K-selected species are generally larger with high adult survival rates, delayed maturation, and low reproductive capacity. A series of mathematical models can quantify a species population's growth in the absence of an interspecific effect. Several mathematical models are introduced to capture the nonlinear density-dependent, having an association with species life-history strategies. Nowadays, a common approach is to use the generalized theta-logistic model as it provides several shapes of density dependence, depending on life-history strategies (Gilpin and Ayala, 1973;Sibly et al., 2005;Williams, 2013). The theta logistic model is discribed as (Williams, 2013) where X is the population density, r max is the maximum reproduction rate, and θ is the nonlinear effect of con-specific density on the growth rate. r max is generally higher for r-selected species as they are high reproductive than K-selected species. The parameter θ controls the shape of the growth curve, which has different biological meanings for r and K selected species. For r selected species, the parameter θ is generally less than one, by the fact that r-selected species are theoretically expected to exhibit a pattern of strong density-dependent reduction in observed r at smaller populations. This happens due to larger consumption of resources by the increased reproductive output, low survivability of offspring, and low parental care (Williams, 2013;Rushton, 1985). In the case of K selected species, it is expected that θ should be greater than one because of the pattern of weak density-dependent reduction in observed r at smaller populations due to low reproductive output, high survivability and high parental care (Williams, 2013;Rushton, 1985) .
We can assume that the growth profile for an N-species community, whose dynamics are driven by competitive species interactions, can follow the theta-logistic growth in the absence of neighboring competitors. This equation can be written as Again in the presence of competitors (species X j , j = 1, ..., N, j = i), the negative effect on the growth of species X i can be written as Thus the competitive system is defined using the following ordinary differential equations where X i is the abundance, r i > 0 is the intrinsic growth rate, and K i is the carrying capacity of the species i(i = 1, ..., N ). θ i is the parameter controlling the shape of the density-dependent growth curve of the i-th species θ i < 1(θ i > 1) for r-selected (K-selected) species. If θ i = 1, the above model is equivalent to the logistic growth equation. Here a ij (< 0) is the competitive interaction coefficient between species i and j.
For a random model of community structure, a link between any pair of species i, j(i, j = 1, ..., N ) occur with probability C (connectance) and the maximum number of links L max is given Newman, 2018). In general, the food web is composed of a hierarchy structure, while mutualism is formed in a bipartite manner. However, there are very few restrictions on a competitive structure. In the natural ecosystem, any structure may be possible for a competitive network. In addition to the random network, we investigate two other types of network structures, viz. scale-free and small-world networks. In a scale-free model, most species have few competitive partners, but some species interact with large numbers of species. The degree distribution of the network nodes follows the power-law P (k) ∝ k −γ , where the probability of finding a node of degree k is P (k), γ is the power-law exponent (Barabási and Albert, 1999;Newman, 2018). A small-world community structure can be viewed as the interpolation of regular and random structures. Regular networks consist of interactions among neighboring nodes with an equal degree of each node, which is then rewired with probability q between randomly selected nodes. If q = 0, it represents a completely regular network, while q = 1 gives a random network. The small-world network is somewhere 0 < q < 1, q being sufficiently small, maintaining a high clustering coefficient and low average path length (Watts and Strogatz, 1998).

Analytical solution for local stability
Analytic behaviors of a system can easily be assessed by linearizing the system near a feasible equilibrium point. The Jacobian matrix corresponding to this simplified system is also known as the community matrix (M). The diagonal elements of the community matrix M can be obtained as partial derivatives of the growth equation of species i with respect to the abundance of itself at equilibrium. Hence for the equation 2.4 the community matrix at the equilibrium point is given by, (2.5) X * i denotes the equilibrium density of the species i (X i > 0). The off-diagonal elements are also found similarly. The off-diagonal element M ij can be obtained as the partial derivative of the growth equation 2.4 of species i with respect to the abundance of species j at equilibrium, i.e., This system is locally stable if all eigenvalues of the community matrix have a negative real part.
The analytical condition of local stability for a small system can be easily obtained by finding the general expression of fixed point for a given parameter space. When the system is too large, we cannot analyze the system through this method. Moreover, it is hard for a given parameter set to determine the fixed points in the presence of the nonlinear density-dependent parameter in the per capita growth rate. Hence, we consider the random matrix theory (RMT) approach, introduced by May in 1972 (May, 1972). In this seminal article, the author considered a random community matrix M of size N × N , which is the Jacobian at the equilibrium point of a random ecosystem containing N species. The element M ij of the matrix M indicates the effect of species j on species i at the equilibrium point. The author assumes that the diagonal elements of the community matrix M are all −1, and the off-diagonal elements follow the normal distribution N (0, σ 2 ), with probability C and 0 with probability 1 − C. For a large number of species, he justified that the system is likely to be unstable whenever the "complexity" K = σ √ N C > 1. On the contrary, some authors considered linkage density (C) and species richness (N ) as a measure of complexity (Pimm, 1984;Landi et al., 2018). Although his work is based on an arbitrary ecological community, further it was extended for particular interaction types and more complex ecosystems (e.g., prey-predator, competitive, mutualistic) by Allesina and Tang (Allesina and Tang, 2012) with the RMT approach. They found remarkable differences between the prey-predator and competitive-mutualistic interactions, the stability increases for the prey-predator interactions and decreases for the other two cases. A substantial number of researchers applied the RMT approach to analyze a random ecosystem's stability in their subsequent studies (Kawatsu and Kondoh, 2018;Kondoh, 2012, 2014;Barabás et al., 2017).
We consider a random competitive community matrix M with connectance C, the elements M ij are taken from half negative distribution −|X|, where E(X) = 0, V ar(X) = σ 2 . Then, (Allesina and Tang, 2012) . To simplify the parameters and species abundance, we set K i = 1, X * i = X * , θ i = θ 1 , θ 2 (for less than one and grater than one respectively) and M ii = d 1 , d 2 (for θ 1 and θ 2 respectively) with probabilistic proportions N P and N (1 − P ) respectively. Note that, for the local stability criteria, the expected spectral abscissa, η(M ), should be in the negative side of the complex plane. The expression of η(M ) (The derivation of the expression is given in the appendix) is given by ( 2.7) for every 0 ≤ P ≤ 1 and x satisfy Resilience is one of the important metrics related to community stability. It measures how fast a locally stable system returns to its equilibrium after a small perturbation. It is calculated as the absolute value of the largest real part of the jacobian matrices M of a stable system at equilibrium, i.e. resilience=|max{Reλ i (M ) : i = 1, 2, ..., N }|. We also provide a measurement of resilience through numerical simulation.

Numerical simulations and Results
We adopt the simulation approach from the study conducted by Chen (Chen and Cohen, 2001), and the operational definition for stability is the same as in that study. We construct random model communities where each species i has probability C(≤ 1, connectance) to interact with species j. We exclude the structures with isolated nodes as isolated species do not exist in nature. To investigate the effect of the density of links and density-dependence variation under more realistic situations, we perform numerical simulations with connectance (C) from 0.1 to 1, by step of 0.1.
We also increase the number of species from 10 to 100 with step-length 10. For simplicity, we take K i = 1. For each structure, we consider hypothetical sets of parameter values from the literature.
The interaction coefficients a ij , a ji are also chosen in the spirit of Chen et.al (Chen and Cohen, 2001). In our model, we deal with the competitive species only, so we are bound to choose the values of the coefficients in the range [-1,0]. For the simulation study, we have chosen interaction coefficients from U[-1,0]. In the simulation procedure, the range for theta is taken from 0.25 to 4, with 0.25 as the step-length (Williams, 2013). It is a known fact that theta must be less than 1 (>1) for r-selected (k-selected) species. To perform numerical simulation, we divide the range of theta (William 2013) into two parts. Any random choices of theta in 0.1 to 1 and 1-4 can be classified to distinguish r and k-selected species. We have chosen the random values of theta from U[0.1,1] and U[1,4], respectively, for identifying r and k-selected species.
For a fixed structure, with particular parameter sets and fixed proportion of species, we generate a positive vector of equilibrium In this procedure, intrinsic growth rates is expected to be large for the specie with θ < 1 with comparison to θ > 1.
For each configuration of species number (N ), connectance (C), and proportion of r-selected species (p), we generate 10000 replicate systems of equation with random parameter sets to estimate the frequencies of a locally asymptotically stable system. The probability of local asymptotic stability is estimated through the relative frequency of locally asymptotically stable systems across all 10000 sample competitive webs for a particular combination of the parameters N , C, and p.
The mean resilience is calculated over the set of locally asymptotically stable systems among these 10000 sample communities for each combination of N, C, and p. We also set the mean resilience to zero for the local stability system, where the frequency is zero. Due to instability, the perturbed system can not reach equilibrium after a sufficiently long time.

Spectral distribution and Spectral abscissa of random community matrices
For an ensemble of randomly generated community matrices with fixed connectance(C) and community size(N ), the expected maximum eigenvalue is given by η(M ) (expected spectral abscissa ), and for stability in the underlying topology, η(M ) should be negative. We investigate how η(M ) is regulated by connectance, community size, and proportion of r-selection. The expected spectral abscissa decreases as the proportion of r-selected species (p) increases in the system for randomly generated competitive community matrices, depicted in the figure 1. The spectral abscissa flips the sign from positive to negative for sufficient proportion, making the system stable. We verify the result with numerical simulation (figure 1) by considering the maximum eigenvalue among fifty randomly generated community matrices for fixed connectance (C=0.5) and community size (N=100).
In figure 2, we illustrated how the proportion of r-selection (p) effect the eigenvalue distribution of random community structures. In the systems with all species either r or K-selected (i.e., p = 0 or 1), the eigenvalue distribution appears to be elliptic. For p lying between zero and one (0 < p < 1), eigenvalue distribution is subdivided into two ellipses, and the sizes of these ellipses are proportional to corresponding proportions of r and K selection (i.e., p and 1 − p respectively) in the system.
As the size of the ellipse close to zero increases, the length of the major axis increases, leading to eigenvalues bringing close to zero and reducing its stability.

Numerical simulations for community Stability and resilience
We nonetheless perform simulation analysis relaxing the assumptions of fixed equilibrium abundance and fixed growth rates for the system. We find that the probability of local asymptotic stability increases monotonically with the proportion of r-selection (p) in the system for fixed connectance as well for fixed community size, for all connectance and community size ( figure 3(a)).
Connectance has been found to have a positive effect on stability (figure 3(a)) as stability increases with connectance for a system with fixed p and community size (N ). But the effect of community size is opposite of connectance ( figure 3(b)), increasing community size decreases stability for the systems with fixed connectance (C) and proportion (p). For the maximum ratio of r selection (i.e., for p = 1) and fixed connectance, systems are found to be certainly stable for any community size. Figure 4 reveals that the proportion of r-selection (p) increases resilience, which means an asymptotic stable system with more r-selected species absorbs small perturbation from the equilibrium more quickly. In the case of systems with all r selected species and different community sizes may stable, but their resilience may differ according to their community sizes (figure 4(b)) In the case of different community size systems with all species r selected may stable but their absorption rate may differ according to community size ( figure 4(b)). Although community size decreases community stability, community size dramatically increases resilience after some threshold proportion for stable systems.
Further figure 5(b) demonstrates the stability complexity relationships for a different proportion of r-selection. Strong stabilizing or destabilizing effects of network complexity emerge at a high proportion of r-selected species. At a low proportion of r-selected species stabilizing effect of the connectance and destabilizing effect of community size investigate. The stabilizing effects are not too strong, at the highest proportion of r-selected (figure 5(d)), and community size act as stabilizing factor at low connectance. Similarly, figure 6(a,c) depicts how the complexity and proportion of r-selected species affect the community's resilience. At a high proportion of r-selection, community size enhances the system's resilience.
We verify the above result for other network types. Figure 7 demonstrates how r-K selection growth affects stability and resilience in scale-free and small-world network structures for various community sizes. For small-world network structure( figure 7(a,c)), stability and resilience similar to the random network case. But the scale-free network ( figure 7(b,d)) exhibits on average low community stability and resilience because a scale-free network generates a low connectance network.
At the highest proportion of r-selected species, the resilience and community stability decreases for increasing community size, which does not contradict the above observation that community size acts as a stabilizing factor for the systems with the highest r-selected species. In a scale-free network, connectance decreases rapidly by increasing community size, and this lower connectance suppresses the stabilizing effect of community size at the highest proportion of r-selected.

Discussion
This study introduces the density-dependent nonlinear growth in community-level dynamics by considering r and K-selection strategies in species growth profiles. We begin our analysis by finding the theoretical condition for stability involving the ratio of r and K-selected species and the complexity of the ecological network for an ensemble of the competitive community. We find that stability increases as the relative proportion of the r-selection increases. The proportion of r-selection in the systems affects stability through eigenvalue push back effect and Jacobian averaging effect (Gravel et al., 2016). We perform numerical simulations to justify our results, and according to our results, competitive webs with more r-selected species are more stable in the sense that the coexistence probability is relatively higher than the systems containing less r-selected species. Note that, Per-capita growth rate plays a major role in this case. As the per-capita maximum growth rate of K-selected species is very low compared to r-selected species, small environmental fluctuation may lead to the extinction of K-selected species, which eventually decreases the stability of the whole community. On the contrary, r-selected species have higher reproduction ability compared to that of the K-selected species. So they can survive in harsh environments and can also handle environmental fluctuations with the help of their high reproductive success (Chaplin III, 1994;Kozlowski, 2012). Also, K-selected species have the higher inter-specific competitive ability. So in a system with more K-selected species, any species is expected to have more competitive pressure than a system with more r-selected species. As a result, stability decreases (Borchert, 1950;Wiens, 1992). It is one of the main reasons why the stability of a competitive community increases with the proportion of r-selected species.
A complexity-stability relationship emerged by varying species richness and density of interaction links. A pattern of stabilization effect of linkage density and destabilization effect of species richness appears by the varying proportion of r and K-selected in a broad range. The stabilization effect of linkage density becomes weak, and the destabilization effect of species richness becomes strong as the proportion of the characteristic of r-selections decreases. This effect of community size and connectance emerged in previous analysis in Mougi (Mougi and Kondoh, 2014), but the seminal analysis of fowler (Fowler, 2009) found that species richness imposes a stabilization effect in the competitive community. When the species growth equation involves a nonlinear term, stability may not increase linearly with species richness (Kawatsu and Kondoh, 2018). In our analysis, stabilizing effect of community size can be found at a high proportion of r-selected, which indicates that the stabilizing effect of community size is context-dependent.
Ecological systems are prone to short-term continuous perturbation. The resilience of a community indicates the time taken by the system to return to the original equilibrium after small perturbations and hence considered as another measure of the stability of an ecological community(Chen and Cohen, 2001)(also known as recovery time). So measurement of the resilience of a system is an important analysis to understand the stability of a system. We conclude from our study that there is a positive relationship of resilience with the richness of the r-selected species of a system. As the r-selected species are highly reproductive, they can recover any moderate reduction of the abundance with the help of their high reproductive ability in a shorter period and go back to the equilibrium abundance faster compared to the K-selected species (Krebs and Davies, 2009). This may be the possible reason for the strong positive correlation of the proportion of the r-selected species in a community with resilience.
In our study, we have considered three types of network topology to validate the robustness of our result. Here we intuitively describe the role of these three network types in the aspect of a competitive community. In a random network structure, any species have an equal probability of competing with others in the community. So the competitive links between species are generated randomly with some specified probability (connectance) (Chen and Cohen, 2001). A large number of species may have a small number of competitors, and a small number of species have many competitors, which can be assumed as specialist and generalist species, respectively. Usually, generalist species have a broader niche width. So they share their niche and form competitive links with more species compared to the specialist species, which have a relatively narrower niche width (May and Mac Arthur, 1972). On account of this, we considered a scale-free network structure, where most of the nodes have few links, and very few nodes have many connections. Lastly, we consider small world networks also have a high clustering coefficient (Newman, 2018). The clustering coefficient of a node indicates how closely the neighbors of a node are connected. Now, if two species have a common competitor, then it is most likely that they will also compete with each other. Our results are consistent with all the above network architectures. Density-dependent self-regulation is one of the essential key features for the stability of complex ecosystems. Barabas et al. (Barabás et al., 2017) pointed out that a food web with insufficiently strong self-regulation cannot be stable. However, strong self-regulation in the growth of r-selected species occurs at low population density and high density for K-selected species. The expected maximum growth of K-selected is low compared to r-selected species. Any self-regulation in the growth of K-selected species enhances the extinction threat.
In all previous studies of ecological networks, including competitive networks, the species per capita growth rate is often assumed to be linearly decreasing with population size (Rozdilsky and Stone, 2001;Fowler, 2009;Mougi and Kondoh, 2014). Still, there is growing usage of nonlinear density dependence population growth in single species model without interspecific competition (Bhowmick et al., 2015;Saha et al., 2013) . Here we emphasize the importance of taking the nonlinear densitydependent growth profile into account on a wide scale. Nonlinear density-dependent growth is widely studied in ecological literature for single-species dynamics (Fowler, 1981;Sibly et al., 2005).
Further, it arises earlier in the competitive community theory in an experiment related to two species of Drosophila (Gilpin and Ayala, 1973), where it suggests that a strong density-dependent in low population may be found for invertebrate species and weak density dependent on the low population for vertebrate species.Our studied model is closely related to that model. It has been hypothesized that an r-selected species has poor competitive ability in a crowded environment while K-selected species have strong competitive ability (Grover et al., 1997). However, analysis of excessive data set reveals that r-selected dominated among all taxa (Sibly et al., 2005). This might be due to the destabilizing role of K-selected species in the stability. Most of the species choose r-selected growth, though they have poor con-specific competitive ability in a crowded environment.
Our study shows a positive relationship between resilience and the richness of the r-selected species of a system. As the r-selected species are highly reproductive, they can recover any moderate reduction of the biomass with the help of their high reproductive ability in a shorter time span and go back to the equilibrium biomass faster than the K-selected species. This may be the possible reason for the strong positive correlation of the proportion of the r-selected species in a community with resilience.
In the two species model, it has been shown that the density-dependent intra-specific effect is more than the inter-specific effect for the coexistence of species (Chesson, 2013). A recent theoretical study also revealed the consequences of nonlinear density-dependent inter-specific interactions in ecological communities (Kawatsu and Kondoh, 2018). Although the consequences according to the nonlinear density-dependent intra-specific effect in the large competitive community were unexplored. We found that nonlinear inter-specific growth has a substantial impact on the stability of a community. In the long-debated factors maintaining the ecosystem's diversity since May's (May, 1972) seminal paper, our analysis adds a new hypothesis to maintain natural communities. Also, we develop community stability criteria for interacting species in a competitive network where species may have different self-regulating effects.

Conclusion
Understanding the functional form of density dependence associated with life-history strategies and its interaction with other species is crucial to shaping our ability to predict ecosystem stability more precisely. The work presented here considers the contribution of r and K-selected species in the stability of the competitive community. One strategy may better respect others in different aspects; for example, in plant communities, r-selected plants get advantages for colonization, whereas Kselected plants benefit from harvesting light, but it was unexplored that how the system's stability effected when we tighten together this species with inter-specific interaction. Ecological networks are prone to a short-term environmental perturbation that may lead to the extinction of one or more species. Interestingly our work reveals that complex competitive communities are stable and absorb perturbations more quickly if the relative ratio of r-selected species is high enough.
The complexity of the ecological community structure also has a significant impact on stability; together with these two life-history strategies, many possible cases emerge in our work. Our derived condition for stability and numerical simulation both represent the same claimed results, and our results contribute to enhancing understanding of how different life-history strategies affect stability. We want to shed light on relatively few empirical studies examining the growth profile in an interactive multi-species context. We also have less information about the growth profile of different life-history strategy species involving interaction with multiple species with varying types of interaction, limiting our prediction. This study can provide clues for the sustainability of complex ecological networks and the maintenance of the population biodiversity by balancing the proportion of r and K-selected species, either adopting natural or artificial strategies like captive breeding or introducing new species in the community. We call for future empirical studies on density-dependent growth profiles to validate our result true in nature.
Local stability of the equilibrium point X * depends on the spectral density abssica η(M ) of M.
The condition for the local stability of the system (1) in the main text is η(M ) should be less than zero.
Then the spectral density abscissa η(M ) is: where δ(.) is the Dirac delta function. In words, any one diagonal entry will be set equal to d 1 with probability P and to d 2 with probability 1 − P . Since D ′ is diagonal, its diagonal entries are also equal to its eigenvalues. We assume that ∆ = d 2 − d 2 > 0 and A ′ = M ′ + D ′ .
We extend it for the case of non-elliptic matrix M , Q is the non diagonal part of M and D ′ is the diagonal matrix. We can write M as, ) 2 + 1 − P x 2 .