THE GLOBAL STABILITY OF COEXISTING EQUILIBRIA FOR THREE MODELS OF MUTUALISM

We analyze the dynamics of three models of mutualism, establishing the global stability of coexisting equilibria by means of Lyapunov’s second method. This further establishes the usefulness of certain Lyapunov functionals of an abstract nature introduced in an earlier paper. As a consequence, it is seen that the use of higher order self-limiting terms cures the shortcomings of Lotka-Volterra mutualisms, preventing unbounded growth and promoting global stability.

Mutualisms usually involve the exchange of resources between species, each of them trading a resource to which it has easy access for another one difficult to acquire.For instance, most terrestrial plants rely on mycorrizal fungi for the uptake of phosphorus, providing carbon in return, while making available nectar and fruits to insects in exchange for pollination and seed transportation, respectively.Other common mutualisms involve protection from natural enemies or from unfavorable environmental conditions.For instance, acacia ants (Pseudomyrmex ferruginea) defend the bull-horn acacia (Acacia cornigera) from browsing by herbivores in exchange for nutrients and sheltering (Palmer et al. [20]).
Mutualisms are of key importance for the apparition of ancient eukaryotic cells, as chloroplasts and mitochondria were initially independent organisms.They are also a major source of new traits, since entire organisms can be acquired, the traits 102 PAUL GEORGESCU, HONG ZHANG AND DANIEL MAXIN gained in this way being more complex than those obtained by transfer of isolated genes (Douglas [3]).Further, all living organisms are currently believed to take part in a mutualistic association at some point in their lives, some in as many as hundreds (Bronstein et al. [2]).
If neither species can survive without the other, as it is the case with the pairing between acacia ants and bull-horn acacia, the mutualism is called obligate, while if each species can survive on their own, the mutualism is called facultative.For instance, a plant that is subject to successful self-pollination or cross-pollination, while also being pollinated by multiple insects, is involved in facultative mutualisms with them.
A central paradigm of modern ecological thinking being the Darwinian theory of natural selection, the antagonistic interactions (predation and competition) received a far greater attention.The fact that the Lotka-Volterra model of mutualism, in which the interspecies competition terms have their sign changed from negative to positive, does normally have unbounded solutions unless the interspecies interaction is weak or asymmetric, probably also helped convey the idea that mutualisms are of less importance.May [17] noted that Lotka-Volterra models are totally inadequate to represent mutualisms, as "they tend to lead to silly solutions in which both populations undergo unbounded exponential growth, in an orgy of mutual benefaction".
To overcome these limitations, and recognizing that mutualisms involve both costs and benefits, such associations between species have begun to be thought as consumer-resource interactions, not unlike predator-prey interactions, but in which benefits outweigh costs (Holland and DeAngelis [13]).In this unifying framework, most mutualisms fit one of the following categories: two-way consumer-resource interactions, one-way consumer-resource interactions and indirect mutualisms via a third species consumer or resource, and also one of the six food web topologies given in [13].Under these assumptions, saturating functions for resource supply and costs associated to providing resources act as limiting factors, preventing unbounded growth.
In Vargas-De-León [24], global stability conditions for two-species models of mutualism are investigated by using suitably constructed Lyapunov functionals and LaSalle's invariance principle.The models of concern in [24] are introduced by Vandermeer and Boucher in [23] to investigate the situation in which the effects of mutualism are density independent and respectively, with regard to the same species x i , i = 1, 2. Also, b 12 and b 21 are strictly positive constants quantifying the mutualistic support the species give to each other.Both models were initially introduced without accounting for the effects of harvesting.Also, if one species is missing, the other behaves in the same way in both models, namely in a logistic fashion.In Georgescu and Zhang [7], paper which attempts to obtain global stability results for mutualisms under more general conditions than those of [24], a general model of a mutualistic interaction is considered: Several sets of hypotheses involving in various proportions monotonicity properties and sign conditions are imposed upon the real continuous functions a 1 , a 2 , f 1 , f 2 , g 1 , g 2 , assuming a priori the existence of a coexisting equilibrium E * = (x * 1 , x * 2 ).Global stability properties for E * under the above-mentioned sets of hypotheses are obtained with the help of the following Lyapunov functionals: .
The functional V 1 , in fact a generalization of the classical Lotka-Volterra logarithmic functional, was first introduced in Harrison [10], although with different signs of the numerators, due to the fact that the interaction considered in [10] is a predation, not a mutualism.See Korobeinikov [16] and Georgescu et al. [5] for the use of functionals related to V 3 to establish the stability of general predator-prey models, Goh [8], Vargas-De-León and Gómez-Alcaraz [26] for stability results for models of mutualism or commensalism by a related approach.See also Korobeinikov [14,15], Georgescu and Hsieh [4], Georgescu and Zhang [6], Vargas-De-León [25], Melnik and Korobeinikov [19], McCluskey [18] for results on the stability of disease propagation models via related functionals.Note that [25] and [6] allow for the possibility of relapse, while [19] allows for age structure and [18] allows for varying infectivity.However, a number of assumptions in [7], although having a certain degree of generality, are rather involved and tailored to the particularities of the models in [24], which they attempt to enlarge.The main results in [7] are given in four theorems.Two of them require strong monotonicity conditions on the components of (3) such as: This last condition is used together with sign conditions on these functions: a 2 g 1 < 0 and a 1 g 2 < 0 on (0, ∞).
The other two theorems use weaker monotonicity conditions but the trade-off is that they each require two extra inequalities, which are necessary for the use of V 3 and V 1 , respectively.These are rather lengthy and involving several auxiliary functions.
Nevertheless, there are biologically relevant situations for which some of these assumptions do not apply.In what follows, we shall prove that the Lyapunov functionals V 1 , V 2 , V 3 are also useful for the study of other models of mutualism, namely for a model with restricted growth rates proposed by Graves et al. in [9] and for versions of (1) and (2) featuring a Richards growth function instead of the classical logistic one.As a byproduct, we derive the fact that a self-limiting term of a higher order prevents unbounded growth and promotes global stability.
2. A mutualistic model with restricted growth rates.In this section, we shall employ the Lyapunov functional V 3 in order to establish the global stability of the coexisting equilibrium for the following model with restricted growth rates proposed by Graves et al. in [9].Named by its proponents "limited per capita growth rate (LGR) mutualism model", the model (6) assumes that each species enhances the growth rate of the other without affecting its carrying capacity.The key assumption made in [9] to derive (6) is that the marginal rate of change for the per capita growth rate of each species due to the increase of the other species is proportional to the difference between the maximum growth rate and the current growth rate.Sufficient biological evidence exists for this modeling assumption such as symbioses between plants and nitrogen-fixing microorganisms (see a detailed description in [9]).
We assume here that r 1 , c 1 , α 1 , r 2 , c 2 , α 2 > 0, that is, the model (6) describes a facultative mutualism (although, mutatis mutandis, our approach can accommodate obligate mutualisms as well) and, in the absence of a species, the growth of the other is of a logistic nature.Let us observe first that (6) fits the abstract framework (3), with A quick review of the assumptions used in [7] shows that none of the stability theorems there can be applied for (6).For example assumption (4) no longer holds.Furthermore, the condition on a 2 g 1 and a 1 g 2 to be negative on (0, ∞) cannot be satisfied here.This means that the theorem in [7] that uses (5) is also inapplicable.The same is true for the theorems that require the auxiliary inequalities.By a classical positivity argument for Kolmogorov models of population dynamics, it is seen that (0, ∞) × (0, ∞) is an invariant region for the system (6).Let us now establish the existence of a coexisting equilibrium for (6).
Theorem 2.1.The system (6) has a unique coexisting equilibrium E * = (x * 1 , x * 2 ).Proof.First, it is seen that the coordinates x * 1 , x * 2 of the coexisting equilibrium E * should satisfy the following relations which lead to = +∞ and ϕ is strictly increasing on [0, ∞), which establishes the existence and uniqueness of a strictly positive root of (8).Consequently, the system (6) has a unique coexisting equilibrium E * = (x * 1 , x * 2 ).After having proved the existence and uniqueness of the coexisting equilibrium, we are now ready to discuss its stability.The first hurdle is the proper choice of a Lyapunov functional.In our settings, V 2 is unusable, due to the terms 1 a1 and 1 a2 , which will not simplify or be everywhere defined.Also, the line of thought involving V 1 , perhaps the simplest of all three Lyapunov functionals, requires a particular inequality between parameters to hold, namely α 1 α 2 ≤ r1r2 c1c2 .We are now left with using V 3 .
We first note that, in this situation, V 3 takes the following form Using V 3 , we may establish the following global stability result, whose proof is given in Appendix A.
Formally, it would have been possible to write (6) in the following equivalent form , which would have led to a simpler expression of V 3 .However, in this equivalent form, g 1 and g 2 have negative signs, while V 3 is constructed for positive g 1 and g 2 , requirement which (6) does fulfill, but (10) does not.Also, the monotonicity of the functions defined in ( 19) is lost using this equivalent form.Actually, this seemingly simpler form fails to meet the conditions for the use of any of the three functionals 3. Mutualistic models with generalized logistic growth.A modification of the classical logistic (or Verhulst-Pearl) equation, in the form p has been proposed by Richards [22] in 1959 in order to better fit the growth of certain biological populations.In the above, x = x(t) represents the size of the population at time t, r is the intrinsic growth rate of the population, K is its carrying capacity and p is an additional parameter which allows for further flexibility in the formulation of the model, specifically for the proper placement of the inflection point (the point where the growth rate is maximal).Apart from its initial purpose, Richards equation has been found useful, among other uses, to predict the severity of a disease outbreak (Hsieh [11]), for which the inflection point represents the turning point of the outbreak and the growth of the basal area of trees (Pienaar and Turnbull [21]).We now attempt to treat versions of ( 1) and ( 2) featuring generalized logistic (Richards) growth, namely models and respectively Let us assume that 0 ≤ e 1 < r 1 , 0 ≤ e 2 < r 2 and that b 12 , b 21 > 0, K 1 , K 2 > 0, observing that for p = 1 the models (11) and (12) reduce to (1) and (2), respectively.For the sake of a better correspondence with the abstract framework (3), we shall denote the coordinates of the positive equilibria for (11) and ( 12) with x * 1 and x * 2 as well, since it will always be clear from the context which model (and therefore which equilibrium) is of concern.
As with the model from the previous section, notice that condition (4) no longer applies.For example, for model (11), with x 1 we can see that a1 f1 +g 2 is not decreasing for p > 1.A similar argument can be made about (12).Denoting we can see that a1 f1 + g 2 is not decreasing on its domain if p > 1.In this last case, condition (5) is also not satisfied.
Let us concentrate on the model (11) first.We start with proving the existence of the coexisting equilibrium of (11), the proof being parallel to that of Theorem 2.1.Proof.As seen from ( 11), the coordinates x * 1 , x * 2 of the positive equilibrium E * 1 should satisfy the following relations which altogether lead to the equation For p = 1, it is seen that ( 14) reduces to 1 is also uniquely defined by , and ϕ 1 is strictly decreasing on [0, ∞), which establishes the existence and uniqueness of a strictly positive root of (14).Consequently, the system (11) has a unique coexisting equilibrium E * 1 = (x * 1 , x * 2 ) whose coordinates verify the equilibrium relations (13).
The fact that no further existence condition is required for p > 1 does not come as a surprise.If p = 1, the model (11) actually represents a Lotka-Volterra mutualism, which is still prone to exponential growth, since the mutualistic terms r1b12 K1 x 1 x 2 and r2b21 K2 x 1 x 2 are of order 2 with respect to (x 1 , x 2 ), the same as the self-limiting terms r1 K1 x 2 1 and r2 K2 x 2 2 .Consequently, a condition limiting the combined strength of the mutualistic interactions is needed in order to prevent unrestricted growth.If p > 1, however, the self-limiting terms are of a higher order, p + 1, than the mutualistic terms, which remain of order 2, and no further precaution is needed.
To prove the global stability of E * 1 , let us first note that inequality which is of great importance in what follows.Note also that if p = 0, the left-hand side is identically 0, while if p > 0 the equality holds if and only if x = 1.Let us observe that (11) fits the abstract framework (3), with In this situation, V 3 takes the form Using again V 3 , we may establish the following global stability result, whose proof is given in Appendix B.
Let us now turn our attention to the study of the model (12).Although the mutualistic interaction acts now in a different way, by decreasing the death rate of each species, rather than increasing the birth rate, as it previously did, one would again expect a conclusion similar to the one obtained for (11) by comparing the orders of magnitude of positive and negative terms in each right-hand side.Namely, if p = 1, the positive and negative terms have equal order and a further condition is needed to prevent exponential growth.If p > 1, the negative terms have a larger order and succeed in limiting the growth of the solutions on their own.
The existence of the coexisting equilibrium can be proved via an argument similar to the one employed for the proof of Theorem 3.1.One then obtains the following result.Let us also observe that (12) fits the abstract framework (3), with In this situation, having in view that 1 time (or, alternatively, V 2 ), one may obtain the following global stability result, whose proof is given in Appendix C. Theorem 3.4.The coexisting equilibrium E * 2 is globally asymptotically stable in (0, ∞) × (0, ∞).
It is to be noted that our Lyapunov functionals can be employed to discuss the stability of certain models of commensalism such as those treated in Vargas-De-León and Gómez-Alcaraz [26], although not without precautions, since, depending on the choices of f 1 , f 2 , g 1 , g 2 , several denominators of V 1 , V 2 , V 3 are null, precluding the respective functional from further use.Also, even after a proper choice of functional, the x 1 -part or the x 2 -part of such functional is null, depending upon which species is not influenced by the other, and should be replaced with a term defined ad hoc.

Conclusions.
Establishing global stability of equilibria in biological models is never a trivial undertaking from a mathematical perspective.Nevertheless, it is highly valuable whenever it is achievable.Most authors rely on local stability theorems for which linearization provides an algorithm that can be applied in all cases (although computational difficulties may arise even in this case).Local stability results do not usually provide a direct information on the basin of attraction of the locally stable equilibrium.From a biological perspective, this is significant because the model is a crude approximation of reality and, if the basin of attraction is small, then a question arises whether the modeled population can realistically be imagined as starting in that basin of attraction to further conclude the convergence toward the equilibrium.
While no universally suitable Lyapunov function exists for all models, we have seen that several types of functions proved suitable for this role.The functionals V 1 , V 2 and V 3 described at the beginning of this paper were used in several cases, including in [7] that provides a general mutualistic model.In that generalization, several technical assumptions were inevitable.However, important biological features do not fit the framework of these assumptions (while other certainly do).In this paper we showed that the functionals mentioned before can be used in several other models of mutualsim that do not satisfy all of the assumptions in [7], namely mutualistic models with restricted growth rate and generalized logistic growth.As mentioned before in this paper, both of these models are important from a biological perspective due to either field evidence and/or added realism by avoiding exponential population growth.Thus, our main result consists in expanding the area of usefulness of these functionals in establishing stability results for these types of models.This is by no means exhaustive.There are still many more possible models of mutualism or commensalism that do not fit all of the assumptions described in [7] or in the present paper.Some examples are Allee effects for the part of the model that describes the independent growth rate of each species or more complex functional forms for the interaction part between the two.Incorporating these assumptions will open research questions not only concerning stability results but also interesting biological ones such as: under what conditions mutualistic or commensalistic species can avoid or lower the Allee threshold which is an interesting question to ask regarding endangered species.This research is currently underway and we will report it in a future article.
Appendix A. Proof of Theorem 2.2.
Proof.First, it is seen that V 3 increases whenever any of |x 1 − x * 1 | and |x 2 − x * 2 | increases and that V 3 (x 1 , x 2 ) ≥ 0, with equality if and only if x 1 = x * 1 and x 2 = x * 2 .Also, the level sets of V 3 do not have limit points on the boundary of (0, ∞) × (0, ∞) since V 3 (x 1 , x 2 ) tends to ∞ if either x 1 or x 2 tends to 0 or to ∞.
We now evaluate the derivative of V 3 along the solutions of ( 6).Our approach is to split • V 3 into two categories of terms, namely terms that can be evaluated by means of monotonicity of certain auxiliary functions Φ 1 and Φ 2 to be introduced below and terms that can be evaluated by means of AM-GM inequality.To this purpose, two equilibrium conditions derived from (7) will be of great importance.One sees that Let us observe that, by algebraic manipulations Note that T 11 ≤ 0, with equality if and only if x 1 = x * 1 .Also, from the equilibrium relations (7), it follows that Consequently, by substituting ( 18) into (17), Let us now define x .
By a similar argument, one may evaluate the second term T 2 in the right-hand side of ( 16) and find that Note that T 21 ≤ 0, with equality if and only if x 2 = x * 2 .Also, from the equilibrium relations (7), it follows that Consequently, It follows that with T 221 ≤ 0, the equality holding if and only if x 2 = x * 2 , since Φ 2 is strictly decreasing.Again, the sign of T 222 is undetermined.
By using ( 16), ( 17) and (20), it now follows that by AM-GM inequality (i.e. the inequality between the arithmetic mean and the geometric mean for two positive numbers), with equality if and only if Since the previous remarks yield that T 11 + T 121 ≤ 0 with equality if and only if x 1 = x * 1 and T 21 + T 221 ≤ 0 with equality if and only if x 2 = x * 2 , we have V 3 ≤ 0, with equality if and only if Proof.First, it is seen that V 3 increases whenever any of |x 1 − x * 1 | and |x 2 − x * 2 | increases and that V 3 (x 1 , x 2 ) ≥ 0, with equality if and only if x 1 = x * 1 and x 2 = x * 2 .Also, the level sets of V 3 do not have limit points on the boundary of (0, ∞) × (0, ∞) since V 3 (x 1 , x 2 ) tends to ∞ if either x 1 or x 2 tends to 0 or to ∞.
We now evaluate the derivative of V 3 along the solutions of (11).We shall split • V 3 into terms that can be estimated by means of (15) and terms that can be estimated by means of AM-GM inequality.To this purpose, we shall also use equilibrium conditions derived from (13).One sees that We note that the equilibrium conditions (13) imply that Consequently, by substituting ( 23) into (22), one obtains by rearranging terms that By (15), T 1 ≤ 0 and T 2 ≤ 0, with inequality if and only if x 1 = x * 1 and x 2 = x * 2 , respectively.We now establish the sign of T 3 + T 4 .It is seen that since the first two terms inside the square brackets are negative by (15) (this is where restriction p ≥ 1 comes from), being equal to 0 if and only if x 1 = x * 1 and x 2 = x * 2 (if p > 1) and being identically 0 if p = 1, while the third one is negative by AM-GM inequality, being equal to 0 if and only if .Also, the level sets of V 1 do not have limit points on the boundary of (0, ∞) × (0, ∞) since V 1 (x 1 , x 2 ) tends to ∞ if either x 1 or x 2 tends to 0 or to ∞.
We now evaluate the derivative of V 1 along the solutions of (12).We shall split • V 1 into terms that can be estimated by means of (15) and terms that can be estimated by means of AM-GM inequality.To this purpose, we shall also use reformulations of the equilibrium conditions.One sees that Note that the equilibrium relations imply that Consequently, by substituting (25) into (24), one may obtain by rearranging terms that By (15), T 1 ≤ 0 and T 2 ≤ 0, with equality if and only if x 1 = x * 1 and x 2 = x * 2 , respectively.We note also that since the first two terms are negative by (15), while the third one is negative by AM-GM inequality.It then follows that T 3 + T 4 ≤ 0, and consequently 2 is globally asymptotically stable in (0, ∞) × (0, ∞).
We shall now observe that a similar conclusion can be reached by using V 2 instead of V 1 .To this purpose, let us note that in this situation V 2 is given by The evalution of the derivative of V 2 along the solutions of (12) gives Using (25), it is seen that A 1 (K By (15), it is seen that T 1 ≤ 0 and T 2 ≤ 0, with inequality if and only if x 1 = x * where

2 .
(15)5), it follows that E 1 ≤ 0, with equality if and only if x 1 = x * 1 and x 2 = x * (if p = 1).Since the previous remarks yield that T 1 ≤ 0 with equality if and only ifx 1 = x * 1 , T 2 ≤0 with equality if and only if x 2 = x * 2 , and T 3 + T 4 ≤ 0, with equality if and only if x 1 = x * 1 and x 2 = x * 2 (if p > 1) and if and only if x1 ≤ 0, with equality if and only if x 1 = x * 1 and x 2 = x * 2 .The rest of the argument is similar to the one displayed above, establishing once again the global stability of E * 2 .