Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains
Abstract
The existence of phenotypic heterogeneity in singlespecies bacterial biofilms is wellestablished in the published literature. However, the modeling of population dynamics in biofilms from the viewpoint of social interactions, i.e. interplay between heterotypic strains, and the analysis of this kind using control theory are not addressed significantly. Therefore, in this paper, we theoretically analyze the population dynamics model in microbial biofilms with nonparticipating strains (coexisting with public goods producers and nonproducers) in the context of evolutionary game theory and nonlinear dynamics. Our analysis of the replicator dynamics model is twofold: first without the inclusion of spatial pattern, and second with the consideration of degree of assortment. In the first case, Lyapunov stability analysis of the stable equilibrium point of the proposed replicator system determines (‘full dominance of cooperators’) as a global asymptotic stable equilibrium whenever the return exceeds the metabolic cost of cooperation. Hence, the global asymptotic stable nature of in the context of nonconsideration of spatial pattern helps to justify mathematically the adversity in the eradication of “cooperative enterprise" that is an infectious biofilm. In the second case, we found nonexistence of global asymptotic stability in the system, and it unveils two additional phenomena  bistability and coexistence. In this context, two inequality conditions are derived for the ‘full dominance of cooperators’ and coexistence. Therefore, the inclusion of spatial pattern in biofilms with noncompeting strains intends conditional dominance of pathogenic (with respect to the hosts) public goods producers which can be an effective strategy towards the control of an infectious biofilm with the drugdependent regulation of degree of segregation. Furthermore, the simulation results of the proposed dynamics for both the discussed scenario confirm the results of the analysis of equilibrium points. The proposed stability analysis not only demonstrate a mathematical framework to analyze the population dynamics in biofilms but also gives a clue to control an infectious biofilm, where phenotypic and spatial heterogeneity exist.
Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains
Jeet Banerjee, Tanvi Ranjan, Ritwik Kumar Layek 
Systems Biology and Computer Vision Laboratory, Dept. of Electronics & Electrical Communication Engineering 
Indian Institute of Technology Kharagpur, India 721302 
{jeetbanerjee, ritwik}@ece.iitkgp.ernet.in, tanviranjan@iitkgp.ac.in 
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\end@floatCategories and Subject Descriptors J.3 [Life and Medical Sciences]: Biology and genetics, Health; J.4 [Social and Behavioral Sciences]: Sociology; I.6.4 [Model Validation and Analysis]

Theory

Biofilm, Evolutionary game theory, Nonlinear dynamics, Population dynamics, Public goods, Social interactions, Spatial pattern
The previous idea of the unicellular behavior of microbes has been metamorphosed into the collective behavioral property [?, ?]. In a natural habitat, bacteria commonly live in a surfaceassociated, matrixenclosed (matrix of extracellular polymeric substances, EPS), complex differentiated communities, called biofilms [?]. A plethora of cooperative processes, such as fruiting body formation [?], summative secretion of EPS [?], biosurfactants [?], virulence factors [?], etc. is required for a microbial community to remain viable. These collective phenomena are actively controlled and regulated by quorum sensing [?, ?] within a subpopulation of public goods producers, called cooperators [?]. However, phenotypic heterogeneity among clonemates is often observed in biofilms [?]. This heterogeneity primarily controls the ability of constituent population to produce and use public goods. Cooperators produce public goods at a cost to themselves, whereas defective strains (defectors) use the produced goods without contribution to the shared pool [?]. It is naturally expected that defectors will thrive under evolutionary pressure due to the low cost of survival; however, the maintenance of cooperation in the colony is supported by the different mechanisms as discussed in the existing literature [?, ?, ?]. Spatial pattern formation is one of the existing mechanisms towards the persistence of cooperation in biofilms [?, ?, ?]. Although, most of these hypotheses suggest that the synergistic and antagonistic interactions within and inbetween of public goods producers and nonproducers primarily regulates the stability of biofilms [?].
Population dynamics models depend on retention of public goods and metabolic cost of public goods production are discussed in [?, ?, ?] for ecological public goods games with constant size and only with cooperative and defective strains. However, a bacterial colony comprises other strains than cooperators and defectors [?]. Moreover, a majority of chronic infections [?] are due to pathogenic biofilms [?, ?]. Therefore, the stability analysis of such biofilms is an important question to study towards the development of novel antibiofilm drugs [?, ?]. In this paper, we formulate the replicator dynamics model [?, ?] in microbial biofilms with noncompeting strains [?] in two ways  first without consideration of spatial pattern and secondly include the pattern formation. Further, we analyze the stability of the equilibrium points of the proposed replicator system using nonlinear control theory [?].
As a result of stability analysis of the proposed system in the absence of spatial pattern, we obtain the ‘full dominance of cooperators’ (i.e. ) in the colony is only a globally asymptotically stable equilibrium whenever retention factor [?] is greater than the cost of cooperation, whereas, is a saddlenode whenever cost exceeds return. Hence, whenever retention exceeds metabolic cost, the cooperators take over the entire population. On the contrary, analysis of replicator equation with consideration of the level of assortment [?] concludes nonexistence of global stability in the system, and it introduces bistability and coexistence. In this context, we derived the inequality conditions for the ‘sole dominance of producers’ and coexistence. Therefore, the conditional dominance of cooperators in biofilms with noncompeting strains and spatial patterns gives a clue to control pathogenicity via regulation of degree of assortment. To the best of the author’s knowledge, this paper presents the first of its kind stability analysis of a replicator system with and without the inclusion of the degree of segregation in a combined framework of evolutionary game theory (EGT) and nonlinear dynamics.
The organization of the paper is as follows. Section Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains discusses the contributions of our work. Section Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains describes the proposed model, stability analysis of the replicator system without consideration of spatial pattern, and corresponding simulation results. The global asymptotic stability analysis of the equilibrium point using Lyapunov function candidate is also presented in this section. Section Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains presents the stability analysis of replicator equation with the inclusion of spatial pattern. In this section, detailed simulations of the dynamics for four different cases are also discussed. Section Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains concludes the paper with a few inferences & remarks on the proposed analysis and future scope.The interdisciplinary work presented in this paper provides a mathematical structure in the context of evolutionary game theory [?, ?] and nonlinear control [?] for rigorous analysis of the social interactions [?, ?] in both pathogenic and nonpathogenic biofilms. In general, this study yields a framework for analyzing multiorder replicator systems.
Existing literature primarily addresses the problems on population dynamics with an assumption of , where is the population density of the competing strain [?, ?]. However, noncompeting strains (as an example “inactive alleles”  metabolic cost for cooperation, productionreception of quorum molecules is zero [?]) those who do not actively take part in public goods game, can be present in the community [?], hence, in this situation . The current work addresses the existence of nonparticipating strains in ecological public goods games while analyzing the evolution of the population. Incorporation of noncomp
eting strains in the present analysis resembles the actual microbial population in a better way compares to the system with only cooperative and defectives strains [?]. Moreover, the presence of nonparticipating strains in the colony of cooperators and defectors gives a scope for rigorous analysis of a secondorder replicator system in support of understanding the dynamics of individual competing subpopulations separately for different microenvironmental conditions. Whereas, it could be possible that the firstorder replicator system for misses some crucial happenings in the population due to the perfectly converse dynamics between competing strains.
We investigate the stability of replicator equation in general and also with an effect of the spatial pattern. The first analysis presents the global stability of the subpopulation of pathogenic cooperators; hence, it gives a general idea for the adversity in the eradication of pathogenicity. Secondly, analysis with the degree of assortment [?] presents a conditional dominance of cooperators while noncompeting strains exist in the colony, and this system supports bistability and coexistence [?].
Social interactions in biofilms, between homotypic and heterotypic strains, is one of the leading causes of the failure of contemporary antibiofilm drugs towards the eradication of bacterial infections [?, ?, ?]. We claim that the proposed analysis, based on the gametheoretic interactions between competing strains, can contribute in developing a novel antibiotics, which concerns the eradication of pathogenicity with the regulation of cost of cooperation and degree of assortment  two primary factors involve in the proposed EGT model.A subpopulation of cells (cooperators) in biofilms often secretes costly public goods, whereas noncooperative strains (defectors) use the produced goods without contributing to the microbial consortium [?]. Here, and are considered as the fraction of cooperators and defectors, respectively, and both and are nonnegative. For a microbial colony of only with the subpopulation of producers and nonproducers, [?]. If total population size is , the number of cooperators and defectors are and , respectively. However, in this paper, we consider the colony (with constant population size) comprises not only cooperators and defectors but also has some other strains [?, ?], i.e. . The assumptions we have taken here are that the remaining strains (other than producers and nonproducers) in the colony do not contain any public good producers by any means, and the average fitness of the population is primarily due to the fitness of cooperators and defectors as the other strains do not take part (i.e. inactive) in the public good competition [?, ?]. Therefore, here we consider, noncompeting strains do not produce the costly public goods by themselves as well as do not receive the benefit from the public goods produced by the cooperators. In this paper, the stability analysis of the proposed replicator system is discussed in twofold: first, without consideration of assortment level in biofilms as discussed in the following subsection and secondly, Section Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains presents the analysis with the inclusion of degree of assortment.
In this subsection, the steady states of a microbial colony for different microenvironmental conditions are investigated without consideration of spatial pattern. These equilibrium points are calculated from the EGTbased replicator dynamics [?, ?] of cooperators and defectors. The replicator equation [?] for and with can be written as
(1) where and are the average fitness of cooperators and defectors, respectively. is the average fitness of the microbial population in ensemble. The standard replicator equation in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) manifests the fitnessdependent growth of a subpopulation. A subpopulation using strategies with a fitness greater than the average fitness ensures increased growth rate while a subpopulation following strategies with lesser fitness than average diminishes in number [?]. In the proposed stability analysis, we use this replicator system as a framework because the fitnessdependent growth of a subpopulation also supports the bacterial proliferation in public goods competition in microbial biofilms [?].
The fitness (i.e. proliferation rate) of a competing strain primarily depends on the amount of resource acquired in competition and the metabolic cost of public goods production. Hence, the fitness functions [?] in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), and can be defined as(2) where is the metabolic cost of production, is the total contribution by a single cooperator to a common pool with multiplication factor (as suggested for ecological public goods games in [?, ?, ?]), denotes the total population size, and . The expression for resource acquired by a center cooperator is , where is the total number of cooperators in the ensemble. The fraction of producers around the center cooperator is , which gives . Substituting in the expression of gives , which is the amount of resource acquired by a center producer from neighboring cooperators. The net metabolic cost of production towards the cooperator is . Therefore, the fitness of a cooperator is the difference between resource acquired and net metabolic cost of cooperation as denoted in the expression of in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains). The contribution of defectors towards the public goods production is zero, i.e. in the expression of .
For , the solutions of the equation (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) are and , where are the integration constants, and , therefore, defectors dominate when public goods producers get a lesser return (from its own production) than production cost . Conversely, pathogenic (to the hosts) cooperators take over the population whenever return exceeds the cost [?]. Hence, for a system with only cooperators and defectors, there is two stable states and for and , respectively. In a control point of view, eradication of infectious biofilm in a system with [?] implies the switching of stable equilibrium from to (domination of public goods nonproducers) by any external influence (drug). Therefore, it might be a bit easier to control such a system with only cooperative and defective strains. On the contrary, in the proposed system with nonparticipating strains, coexisting with public goods producers and nonproducers, we determined (domination of public goods producers) as only the global stable equilibrium. This global domination of cooperators states a considerable difficulty in the eradication of pathogenic biofilms compares to the system in literature [?, ?] as the cooperators predominantly influence the chronic bacterial infections towards the hosts.
We consider and . Hence, the corresponding fitness functions in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) can be redefined in terms of and as(3) By substituting (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), the modified dynamics is defined as
(4) Fixed points of the replicator system in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) are calculated using , which manifests the equilibrium points as , and a line of fixed points, . The system in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) contains an infinite number of fixed points in the form , which are unstable due to the absence of cooperators [?]. However, we analyze the equilibrium points , and as these are directly obtained from (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains). In the process of evaluating the fixed points from (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), a system of equations can be written as
(5) By solving (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), we can write , which implies as . Moreover, in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) results as and for the system of equations in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains). Hence, the microbial ensemble comprises cooperators and defectors only (no nonparticipating strains) for . In other words, the solution of (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), i.e. is a line of fixed points, which is a special case of (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) for .
The stability of previously calculated fixed points is investigated for the proposed dynamics as defined in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains). The replicator dynamics in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) can be redefined as
(6) The method of linearization using Jacobian in the neighborhood of equilibrium points is used to investigate the stability of the system with respect to the fixed points. The Jacobian matrix, can be expressed from the replicator system in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) as
(7) where
(8) signifies the microbial colony solely comprises noncompeting strains. The Jacobian matrix, for (0,0) is calculated using (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) and (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) as
(9) and , where and denote the trace and determinant of , respectively. For , , and for , . for any relation between and . (, ) analysis of the fixed point represents the borderline case, hence, the nature of cannot be determined directly. Moreover, the eigen values of are = (, ), and and are the corresponding eigen vectors. Here, denotes eigen value of Jacobian, for equilibrium point. For , and for , , therefore, the eigen vector converges to for and diverges from for . However, it is difficult to denote the nature (converging or diverging) of eigen vector for from both (, ) and eigen value analysis of . Hence, for further analysis of the nature of eigen vector , we use centre manifold theory [?, ?]. A parabolic manifold with centre at (0,0) and tangent to can be defined as . In the formulation of centre manifold, the replicator dynamics in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) is denoted as
(10) By comparing (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) and (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), we can write = , = , and = . We use the following expression [?] to calculate in the equation of centre manifold .
(11) By substituting the values of , , and in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), we can get
(12) Neglecting the higher order terms (h.o.ts) in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), we get as for the centre manifold. gives the centre manifold expression as . For , we get the system dynamics from (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) as . Hence, centre manifold is an unstable manifold, and corresponding fixed point (0,0) is an unstable node with respect to the centre manifold . In place of , if we consider the centre manifold as , we can get the expression after neglecting h.o.ts from (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) as . gives the centre manifold equation as , and corresponding dynamics , which is same as previous. For , the eigen vectors converges and diverges, hence, (0,0) can be considered as a saddlenode. But signifies (0,0) is not a saddle for , can be considered as an unstable node. For , both the eigen vectors diverge from (0,0). Hence, (0,0) is an unstable equilibrium point for both and .
The equilibrium point () signifies the absence of cooperators in the colony, and a fraction of defectors coexist with the nonparticipating strains. The Jacobian matrix, at the equilibrium point () is , which is same as . = and . () can be expressed in terms of and as , which is greater than one for , hence, is only a valid equilibrium point for as . For , and . Eigen values of are = (, ). for , hence, corresponding eigen vector diverges from . To determine the nature of eigen vector for , we use centre manifold theory. The corresponding equation of parabolic manifold centered at and tangent to is , where . After neglecting h.o.ts, We can get the expression from (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) for as
(13) L.H.S of (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) is equal to zero if . Hence, the expression of centre manifold , and for , , which manifests diverging eigen vector of . For , eigen vectors ( and , respectively) corresponding to the eigen values diverge from the fixed point, therefore, is an unstable node.
The fixed point implies the sole domination of public goods producers in biofilms. The Jacobian matrix, at the fixed point is
(14) From (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), we can write and . manifests the inequality , which is true as and . Hence, for any relation between and . , which is for and for . , which is always . Therefore, from the analysis, we can ensure that the equilibrium point is a saddle point for and a stable node for . Moreover, eigen values of are = , and corresponding eigen vectors are and . , which is as and . for and for . From the eigen value analysis it is clear that the is a saddlenode for (as , ), and a stable equilibrium for (as ). Hence, both and eigen value analysis of ensure the same nature of for and . The question remains is that is only stable or asymptotically stable equilibrium point for , and if asymptotically stable, then is it locally or globally asymptotically stable? To investigate this question, we analyze the point using Lyapunov function candidate [?] as discussed in the following.
The Lyapunov function candidate, can be defined as
(15) where the state variable = , is a symmetric matrix with principal minors greater than zero, i.e., and . The elements of the matrix can be determined from the following Lyapunov equation [?].
(16) where is the Jacobian matrix at the equilibrium point as defined in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), is a positive definite matrix, considered here as an identity matrix, hence . By solving the Lyapunov equation in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), we can get the values of , and as
(17) Now, the eigen values of are defined as
(18) where , .
and are for as , and for , . Moreover, . As , we can write , hence, , which implies is a positive definite matrix. Lyapunov stability theorem suggests that the system is asymptotically stable () with respect to the equilibrium point for which the matrix is positive definite [?]. In the proposed system, for , hence, at this juncture we can say that the system is locally asymptotically stable with respect to the fixed point for . By substituting (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), can be defined (after coordinate transformation) as(19) From (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) it is clear that and as , and for , . Hence, the necessary conditions for Lyapunov candidate function are satisfied. can be expressed as
(20) The expressions of and as denoted in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) and (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), respectively are simulated for and in Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains. From Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains (c) and (d), it is clear that is radially unbounded [?] and for , respectively. Hence, we can conclude that the equilibrium point is globally asymptotically stable for , whereas is not a stable equilibrium for as and (Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains (a) and (b), respectively).
a b c d Figure \thefigure: (Color online) Simulation of the expressions of and as formulated in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) and (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), respectively for and . For , (a) and (b) . For , (c) and it is radially unbounded and (d) The equilibrium state represents the absence of defectors in the colony, and cooperators coexist with the nonparticipating strains. The Jacobian matrix, at the equilibrium point is
(21) For , , hence, is only a valid fixed point for . implies the inequality , which is true as and , hence, for . , and . Therefore, from analysis, it is clear that the fixed point is an unstable node for . Moreover, eigen values of are . For , both and are , hence, eigen value analysis also ensured that is an unstable equilibrium for .
The fixed state suggests the ‘full dominance of defectors’ in the colony. We analyze the fixed point as it is one of the boundary solutions to . The Jacobian matrix, at is
(22) Eigen values of are , and corresponding eigen vectors are and . for and for . Hence, the eigen vector converges to for , and diverges from the fixed point for . Next, we investigate the nature of eigen vector corresponding to eigen value using centre manifold theory as discussed in subsections Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains and Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains.
The parabolic manifold centered at and tangent to can be considered as . To calculate the value of , we use equation (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), and corresponding expression after neglecting h.o.ts is(23) Equation (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) resembles (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) with . The solution to (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) is , which gives the centre manifold as , and corresponding system dynamics is . Therefore, is an unstable manifold, and diverges from the equilibrium point . Hence, for both and , is an unstable fixed point.
The summary of the stability analysis of fixed points (as discussed in subsections Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains to Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) is tabulated in Table Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains.Fixed points Nature of fixed points (0,0) Unstable node Unstable node Not valid Unstable node (1,0) Saddlenode Globally asymptotically stable node Unstable node Not valid (0,1) Unstable node Unstable node Table \thetable: Summary of stability analysis of fixed points
Phase portraits for and are simulated in Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains. For this simulation, we consider the proposed dynamics as formulated in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains). Though for , all the initial conditions converge to the equilibrium point (figure (a)), for the stability of does not hold true as depicted in figure (b). Hence, for , is an unstable equilibrium. Whereas figures (c) and (d) confirm as globally asymptotically stable equilibrium for both and , hence, is only the globally asymptotically stable fixed point for the proposed dynamical system. The dynamics of the fraction of cooperators and defectors are simulated from (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), and the results are depicted in Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains. Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains clearly states that is globally stable for ((c) and (d)), whereas is an unstable node for ((a) and (b)). The simulation results in Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains and Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains match with the stability analysis of equilibrium points as discussed in subsections Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains to Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains.
Although is a globally asymptotically stable equilibrium for , for , the fixed point is a saddlenode as discussed in the subsection Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains, and corresponding phase portrait (zoomed in the neighborhood of ) is depicted in Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains (a). For , the proposed dynamics in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) converges to , i.e. the microbial colony consists of cooperators and defectors only. The simulated phase portrait for is shown in Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains (b) which ensures the convergence of the system towards .a b c d Figure \thefigure: (Color online) Phase portraits for (a) , (b) , (c) and (d) . and denote initial conditions and equilibrium point(s), respectively. a b c d Figure \thefigure: (Color online) Simulation of dynamics of and from (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains). (a) (, )= (0,1) is the stable equilibrium for , (b) (0,1) is an unstable fixed point for as the fraction of defectors converge to different steady states depending on the initial conditions. (, )= (1,0) is the stable fixed point for (c) and (d) . Legend of figure (a) is applicable to the figures (b), (c) and (d). a b Figure \thefigure: (Color online) (a) The equilibrium point is a saddle for , (b) proposed dynamical system in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) converges to the line for . and denote initial conditions and equilibrium points, respectively. From the previous stability analysis, it is clear that the fixed point is only a globally asymptotically stable equilibrium for for the proposed dynamics in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains). The equilibrium point signifies that the microbial colony is dominated by the public goods producers (so called cooperators), which are primarily responsible for the persistence of microbial infections [?, ?]. Due to the asymptotically stable nature of the equilibrium point , the microbial system has an inherent tendency to occupy the whole consortium by the pathogenic cooperators, and this is the reason for the difficulty in the eradication of infectious biofilms.
In the previous section, we discussed the stability analysis of the equilibrium points of the replicator system in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) using the fitness functions as defined in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains). In this stability analysis, the payoff expressions in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) do not contain any parameter(s) which signify the degree of assortment, present in microbial biofilms. As a result of the previous analysis, a single fixed point (‘full dominance of cooperators’) satisfies the condition for global asymptotic stability. In the present section, we extend the previous discussion via inclusion of degree of assortment in the fitness functions of cooperators and defectors. The purpose of including the spatial pattern in the proposed analysis is twofold: (a) emergence of cooperation in a population with phenotypic heterogeneity facilitates spatial patterns in microbial ecosystems [?], and (b) the degree of assortment can be controlled externally with the variation of initial cell density [?]. Hence, it could be interesting to visualize the effect of level of segregation on the population dynamics of cooperative and defective strains. Segregation in biofilms favors cooperative cell lines as the cooperators more often interact with their lineages, and high level of assortment resists exploitation of public goods by the noncooperative ‘cheaters’ [?, ?]. On the contrary, defectors are advantaged in wellmixed colony. To define the fitness functions with the consideration of spatial patterns, as formulated in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), and are added as an intrinsic advantage to the cooperators and defectors, respectively, and . Therefore, defines segregated biofilms, whereas, ensures wellmixed.
(24) By substituting the payoff expressions (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) in the replicator equation (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), we can write the dynamics of cooperative and defective strains in a segregated or wellmixed colony with nonparticipating strains as
(25) The equilibrium points, , are calculated from (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) using =0 and =0 as , , , , and the point of coexistence = . The dynamics in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) contains infinite number of fixed points in the form where , however, an ecological public goods game with nonparticipating strains (i.e. average fitness of the population is dominated by only the competing strains) does not hold any asymptotically stable equilibrium in the absence of public goods producers [?]. Even though we analyze the fixed points , as these are directly calculated from the dynamics in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains), and we consider as it is a boundary value of the infinite set of equilibrium states. A notable difference between the dynamics in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) and (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) is that the incorporation of the level of assortment induces a unique point of coexistence [?, ?] between the competing strains.
Four possible cases are considered for the analysis of population dynamics in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains)  case I: implies return is less to the cooperators than the metabolic cost of cooperation in a wellmixed biofilm, case II: manifests return is greater than the cost of cooperation in a segregated biofilm, case III: denotes return is less than the meatbolic cost in an assorted biofilm, and case IV: signifies the return is greater than the cooperative cost in a wellmixed biofilm. Population dynamics of cooperators and defectors are simulated from (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) for the case I to case IV in the scenario , and corresponding results are depicted in Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains for a set of parameters , , and .a b c d e Figure \thefigure: (Color online) Simulation of dynamics of and from (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains). (a) Defectors dominate for , ; (b) Cooperators dominate for , ; (c) and (d) The domination of cooperators or defectors depends on the relation between initial fraction and level of coexistence for , ; (e) Coexistence between cooperators and defectors for , . Legend of figure (a) is applicable to the figures (b), (c) and (e).
From Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains (a), it is clear that the lesser return to the cooperators than the metabolic cost of production in wellmixed biofilms (case I) favors defectors, whereas, the greater return than the cost of cooperation in segregated biofilms (case II) supports cooperators as depicted in (b). Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains (c, d) present bistability (winning of cooperators or defectors depend on the initial fraction) when the return is less than the cost of cooperation in assorted biofilms (case III), and Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains (e) illustrates coexistence in a wellmixed colony with the greater return to the cooperators compared to the metabolic cost of public goods production (case IV). However, is not an asymptotically stable equilibrium point for a replicator system with noncompeting strains as discussed previously. Therefore, the stable as shown in Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains (a) and (c) may not be stable for the other set of , even though, that set satisfies the relation for the cases I and III. The point of coexistence is also not a stable state in the entire domain of case IV as shown in Fig. Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains. It is shown that coexistence persist for , however, the dynamics moves to another steady state while and is unchanged. Therefore, the stability conditions of the equilibrium points for the valid cases are necessary to investigate further. Conditions for the stability of the fixed states and are discussed in the following subsection.
a b Figure \thefigure: (Color online) Dynamics of and for the case IV. (a) Coexistence occurs with and (b) Cooperators dominate for . The replicator dynamics in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) can be written alternatively as
(26) In the next subsections, we investigate the stability conditions of the fixed points and using linear stability analysis via calculating the Jacobian from (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains).
The Jacobian matrix of is calculated from the dynamics in (Stability Analysis of Population Dynamics Model in Microbial Biofilms with Nonparticipating Strains) as
(27) The eigen values of are
