Dynamics of a mechanistically derived stoichiometric producer-grazer model

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1 Journal of Biological Dynamics ISSN: Print Online Journal homepage: Dynamics of a mechanistically derived stoichiometric producer-grazer model Hao Wang, Yang Kuang & Irakli Loladze To cite this article: Hao Wang, Yang Kuang & Irakli Loladze 28 Dynamics of a mechanistically derived stoichiometric producer-grazer model, Journal of Biological Dynamics, 2:3, , DOI: 1.18/ To link to this article: Copyright Taylor & Francis Group, LLC Published online: 28 Jan 29. Submit your article to this journal Article views: 29 View related articles Citing articles: 14 View citing articles Full Terms & Conditions of access and use can be found at Download by: [ ] Date: 6 November 217, At: :31

2 Journal of Biological Dynamics Vol. 2, No. 3, July 28, Dynamics of a mechanistically derived stoichiometric producer-grazer model Hao Wang a *, Yang Kuang a and Irakli Loladze b a Department of Mathematics, Arizona State University, Tempe AZ, USA; b Department of Mathematics, University of Nebraska - Lincoln, Lincoln, NE, USA Received 24 July 27; final revision received 13 October 27 One of the simplest predator-prey models that tracks the quantity and the quality of prey is the one proposed by [I. Loladze, Y. Kuang, and J.J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math. Biol pp ]. In it, the ratio of two essential chemical elements, carbon to phosphorus, C:P, represents prey quality. However, that model does not eplicitly track P neither in the prey nor in the media that supports the prey. Here, we etend the by mechanistically deriving and accounting for P in both the prey and the media. Bifurcation diagrams and simulations show that our model behaves similarly to the. However, in the intermediate range of the carrying capacity, especially near the homoclinic bifurcation point for the carrying capacity, quantitative behaviour of our model is different. We analyze positive invariant region and stability of boundary steady states. We show that as the uptake rate of P by producer becomes infinite, s become the limiting case of our model. Furthermore, our model can be readily etended to multiple producers and consumers. Keywords: stoichiometry; Droop model; logistic equation; producer-grazer model; phosphorus uptake AMS Subject Classification: 92D25; 34C6 1. Introduction Ecological stoichiometry is the study of the balance of energy carbon and multiple chemical elements phosphorus, nitrogen, etc. in ecological interactions [11]. All organisms are composed of these elements. The relative abundances of them vary considerably among species and across trophic levels; hence, this chemical heterogeneity among species can have significant effects on food web dynamics. From the stoichiometric point of view, both food quantity and food quality need to be implicitly or eplicitly modeled in producer consumer interactions, since it is observed that the plant quality as well as its quantity can dramatically affect the growth rate of herbivorous grazers and may even lead to their etinction. On the other hand, plants are indeed limited by nutrient availability [3], and herbivores are more nutrient-rich organisms than plants, inducing food quality constraints. Stoichiometric theory has strong potential for both quantitative and qualitative improvements in the predictive power of population ecology [2,1]. *Corresponding author. wanghao@math.gatech.edu ISSN print/issn online 28 Taylor & Francis DOI: 1.18/

3 Journal of Biological Dynamics 287 The rapidly growing empirical study of ecological stoichiometry facilitates a variety of stoichiometric population models in recent years [1,4,5,7,8,9]. Simple phenomenological twodimensional producer consumer models, such as [8,9], are mathematically tractable but some of its simplifying assumptions may not hold in reality. Kuang et al. [7] gave a reasonable interpretation of the Loladze, Kuang and Elser using the Droop equation for the plant growth. This paper is to formulate the stoichiometric plant-herbivore model more mechanistically by eplicitly modelling P content in the prey and in the media, thus relaing a main assumption in the. Our model is a system of four autonomous ordinary differential equations, which can be reduced to a three-dimensional system. To compare our model formulation with the, it is convenient to recall the main assumptions used in the. A1 The total mass of phosphorus in the entire system is fied. A2 Phosphorus to carbon ratio P:C in the plant varies, but it never falls below a minimum q mgp /mgc; the herbivore maintains a constant P:C ratio, denoted by mgp /mgc. A3 Phosphorus in the system is divided into two pools: phosphorus in the herbivore and phosphorus in the plant. 2. Model formulation Let be the density of carbon content in the producer, p be the density of phosphorus content in the producer, y be the density of carbon content in the grazer and P be the density of free phosphorus in the media. Our first two equations are similar to, ecept that the phosphorus density of producer, p, and the free media phosphorus, P, are now eplicitly modeled here. The above Assumption A3 is no longer needed. The new model takes the form of d dy dp dp = r 1 mink,p/q} producer growth limited by nutrient and light = ê min 1, p/ } fy grazer growth limited by food quality and quantity p = gp fy P uptake by producer P loss due to grazing = gp P uptake by producer p + ê min, p } fy. } } P recycling from grazer feces + dp P recycling from producer fy uptake by grazers, 1a ˆdy, grazer death and respiration loss 1b dp, 1c P loss due to producer recycling + ˆdy P recycling from dead grazer 1d The tet below each term eplains its biological meaning. r is the intrinsic growth rate of the producer day 1. ˆd is the specific loss rate of the grazer that include the rates of respiration and death day 1. d is the phosphorus loss rate representing P lost due to leave dropping, root

4 288 H. Wang et al. decomposition and other recycling processes. fis the ingestion rate of the grazers, which will be assumed to follow Holling type II functional response in our analysis and simulation work. In general, fis a bounded smooth function that satisfies the following assumptions: f =, f > and f, for. 2 gp is the phosphorus uptake rate of the producer. This function has similar properties as f. ê is the maimal production efficiency. The second law of thermodynamics requires that ê < 1. K represents a constant carrying capacity related to light. q is the minimal cell quota P : C of the producer. is the fied cell quota of the grazer. q in reality; hence, we assume that >q in this paper. In the free phosphorus equation, gp is the P uptake by the producer, dp and ˆdy are P recycling from producer and consumer death, p ê min, p } fy describes the consumer s P consumption from the producer minus the actual P retained due to growth and maintenance needs, which gives the amount of P recycled from consumer droppings and other losses. T = p + P + y is the total phosphorus of the system, composed of producer phosphorus, free phosphorus and consumer phosphorus. By a simple calculation, we have dt/ =. The total phosphorus of System 1 is kept at the constant level of T = p + P + y. Thus, P T p y, which can be used to reduce System 1 by one dimension to d dy dp = r 1 =ê min 1, p/ mink,p/q} fy, } fy ˆdy, 3a 3b = gt p y p fy dp. 3c The parameters in the P column of Table 1 will be needed for our subsequent study of System 3. Their values in the V column will be used for numerical simulations to be present later. Table 1. The parameters P of System 3 and their values V used for numerical simulations. P Description V Unit r Intrinsic growth rate of the resource.93 day 1 K Resource carrying capacity determined by light.25 2 mg C/1 c Maimal ingestion rate of the grazer.75 day 1 ĉ Maimal phosphorus uptake rate of the producer.2 mg P/mg C/day a Half-saturation constant of the grazer.25 mg C/1 â Phosphorus half-saturation constant of the producer.8 mg P/1 ê Maimal conversion rate of the grazer.74 ˆd Loss rate of the grazer.22 day 1 d Phosphorus loss rate of the producer.5 day 1 Constant P:C of the grazer.4 mg P/mg C q Minimal possible P:C of the producer.4 mg P/mg C T Total phosphorus in the system.3 mg P/1

5 Journal of Biological Dynamics Mathematical analysis In order to have a basic mathematical understanding of Model 3, we would like to study the dissipativity of the system and the stability of its boundary steady states. The following theorem shows that the system is naturally dissipative, meaning that the realistic solutions will remain in a biologically meaningful region. THEOREM 3.1 Solutions with initial conditions in the set =,y,p: <<mink,t/q}, <y, <p,p+ y < T } remain there for all forward times. Proof We consider a solution Xt t, yt, pt of 3 with initial condition in. Hence, < <mink,t/q}, <y, <p, p + y <T. Assume that there is time t 1 > such that Xt touches or crosses the boundary of closure of for the first time, and then t, yt, pt for t<t 1. We will have several cases to consider. Case 1 t 1 = butpt 1 =. pt, pt + yt T for t t 1 implies yt T/ for t t 1. Let p 1 = minpt : t [,t 1 ]} >. Then for t t 1,wehave d = r 1 fy mink,p/q} r 1 mink,t/q} mink,p 1 /q} = [ r 1 mink,t/q} mink,p 1 /q} where μ is a constant. Then, t e μt e μt 1 >, a contradiction. f T / ] f T / μ, for t t 1, which implies that t 1 Case 2 t 1 = mink,t/q}. yt, pt + yt T for t t 1 implies pt T for t t 1. Then for t t 1,wehave d = r 1 fy r 1 mink,p/q}. mink,t/q} The standard comparison argument yields that t < mink,t/q} for all t t 1, a contradiction. Case 3 yt 1 =. Then for t t 1,wehave dy =ê min 1, p/ } fy ˆdy ˆdy. Hence, yt ye ˆ > for t t 1, a contradiction.

6 29 H. Wang et al. Case 4 pt 1 =. pt + yt T for t t 1 implies T pt yt for t t 1. Thus, gt pt yt for t t 1. Since pt, pt + yt T for t t 1, then yt T/ for t t 1. Thus, for t<t 1,wehave dp = gt p y p fy dp p fy dp [ f T / d]p νp, where ν is a constant. Thus, pt pe νt > for t t 1, a contradiction. Case 5 pt 1 + yt 1 = T. Let zt = T pt yt, then zt 1 = and zt > for t<t 1. Then for t t 1,wehave dz = dp dy gz + dp + ˆdy g z mink,t/q}+mind, ˆd}T z = mind, ˆd}T [g mink,t/q}+mind, ˆd}]z μ νz, where μ > and ν > are constant. Thus, zt e νt z > for t t 1, a contradiction. The stability analysis for general forms of fand gp is tedious. For simplicity, we assume below that gp = αp and f= β. In this special case, Model 3 takes the following eplicit form system becomes d dy dp = r 1 =ê min 1, p/ mink,p/q} βy F,y,p, } βy ˆdy yg, y, p, 4a 4b = αt p y βpy dp H,y,p. 4c The boundary equilibria include the total etinction equilibrium E =,, and the grazer etinction equilibrium TK K,,, if K< T K + d/α q d α ; E 1 = T q d T α,,q q d, if K> T α q d α, a.12 b 1 c.9 plant plant grazer grazer Plant and grazer biomass d=5, both plant and grazer go etinct. Plant and grazer biomass d=2, grazer goes etinct while plant survives. Plant and grazer biomass plant grazer d=.5, both plant and grazer survive Time day Time day Time day Figure 1. Figure illustration of Theorems by choosing different d values. a The etinction steady state E is asymptotically stable.b The grazer etinction steady state E 1 is asymptotically stable. c The coeistence steady state E is asymptotically stable.

7 Journal of Biological Dynamics 291 if T/q d/α >. Model 4 may also have a coeistence equilibrium E, which satisfies r 1 = βy, 5a mink,p/q} ê min 1, p/ } β = ˆd, 5b αt p y = pβy + dp. 5c For the etinction steady state E, we have the following theorem for the general system 3. THEOREM 3.2 The etinction steady-state E =,, in System 3 or 4 is globally asymptotically stable if d > mgt, where m = min /p, 1 +[d + f T //r]/q }. Proof Let u = /p, then du = d/ p u dp/ p ru1 qu + d + f yu ru1 + d + f y/r qu ru1 + d + f T //r qu. Hence, u min /p, [1 + d + f T //r]/q } m. For the p equation, dp/ gt dp gt mp dp = gt m dp. Since d > mgt implies gt m d<, hence p ast. Net, we show that ast. This can be seen from the fact that d/ r 1 q/p, which implies that lim sup t t p/q. Clearly, this also implies that y as t. The above theorem states that both the plant and the grazer go etinct when the nutrient loss rate is high. Plant etinction cannot be obtained in most previous models including the, ecept in pure ratio-dependent models [6]. Therefore, our model provides an alternative and possibly less controversial deterministic mechanism to ehibit such total and realistic etinction dynamics. We will compute the Jacobian matri through F, G, H and their partial derivatives. The partial derivatives of these functions are r if p>qk, F = mink,p/q}, F y = β, F p = rq if p<qk, p 2 êβ if p>, if p>, G = G y =, G p = if p<, êβ if p<, H = αt p y, H y = α βp, H p = α βy d. The Jacobian matri is F + F F y F p A = yg G + yg y yg p. H H y H p The trace of A, tra = F + F + G + yg y + H p.

8 292 H. Wang et al. The determinant of A, deta = F + F G + yg y H p + yf y G p H + yf p G H y F p G + yg y H yf y G H p F + F yg p H y = H yf y G p F p G yf p G y + yh y F p G FG p F G p + H p F G + yfg y + F G + yf G y yf y G. Let A jk be the determinant of the 2 2 matri resulted from A by removing the jth row and the kth column. Then, A 11 = G + yg y H p yg p H y, A 22 = F + F H p F p H, A 33 = F + F G + yg y yf y G. By Routh Hurwitz criterion, all eigenvalues of A have strictly negative real parts if the following conditions hold. i tra <; ii deta <; iii deta tra 3 k=1 A kk >. The following theorem presents a sufficient condition for the local stability of E 1 when K> T/q d/α >. THEOREM 3.3 For the case of K>T/q d/α >, E 1 is locally asymptotically stable if êqβ/t /q d/α < ˆd. Proof Recall that E 1 =,, p where = T /q d/α, p = qt /q d/α. Clearly, < K, p < qk. Since > q, we have p <. Observe that U = êqβ/ ˆd <, V = αt r/q rd > and W = α + r + d>by conditions of the theorem and the positivity of parameters. In our case, the Jacobian matri at E 1 becomes r β r/q A = U. αt p α β p α d The U, V and W formulae together with some straightforward calculations yield and tra = U α r d<, deta = UV <, 3 deta tra A kk = WU 2 UW + V>. The theorem follows from an application of the Routh-Hurwitz criterion. k=1 The net theorem presents a sufficient condition for the local stability of E 1 K<T/q d/α > and T<K+ d/α. when THEOREM 3.4 Assume that K < T /q d/α and T /K + d/α<. Then, E 1 is locally asymptotically stable if êβ/t K/K + d/α < ˆd. Proof In this case, E 1 =,, p where = K, p = T K/K + d/α. We see that p >qk, p <. Observe that U = êβ p/ ˆd = êβ/t K/K + d/α ˆd < and V = α d<. By straightforward calculations involving the Jacobian matri, r βk A = U. αt p α β p α d we have tra = U + V r<, deta = ruv < and 3 deta tra A kk = U + V UV ru rv r 2 U r 2 V>. k=1 The theorem follows from an application of the Routh Hurwitz criterion.

9 Journal of Biological Dynamics 293 Table 2. Stability results of E 1 in System 4. Case Condition Stability T q d α K> T q d α > êqβ T d α <K< T q d α êqβ êβ êβ T q d α T q d α TK K + d/α < ˆd TK K + d/α > ˆd Not eist < ˆd LAS > ˆd Unstable LAS Unstable K< T d α êβk < ˆd êβk > ˆd LAS Locally asymptotically stable Unstable Our last theorem presents a sufficient condition for the local stability of E 1 K<T/q d/α > and T>K+ d/α. when THEOREM 3.5 Assume that K < T /q d/α and T /K + d/α >. Then, E 1 is locally asymptotically stable if êβk < ˆd. Proof We see that E 1 =,, p where = K, p = T K/K + d/α. Then, p >qk, p >. Let U =êβ ˆd =êβk ˆd < and V = α d<. By straightforward calculations, we see that the Jacobian matri in this case is the same as that in the previous theorem with the new definition of U. The rest of the proof is identical to that of the previous theorem. Observe that when T /q d/α, E 1 does not eist. E 1 degenerates to E if T /q d/α =. Instability results of E 1 can be easily obtained from the conditions lead to deta >. In addition, > q is biologically reasonable. With this assumption, the condition K < T /q d/α and T /K + d/α < is equivalent to T / d/α<k<t/q d/α, and the condition K < T /q d/α and T /K + d/α > is equivalent to K < T / d/α. We summarize these results on E 1 in Table 2. All stability results in this section are illustrated in Figure Numerical dynamics and its implications The initial conditions in our simulation are chosen inside the biologically meaningful region. We let f= c/a + and gp =ĉp/â + Pin our simulations. Since the carrying capacity K can be linked to or regarded as the light intensity in lab and field eperiments, it is one of our natural bifurcation parameters. In Figure 2, We vary the carrying capacity K in the bifurcation diagrams. Other parameter values are given in Table 1. In this figure, we simulate Model 3 with faster nutrient process and compare the results to those of. In Figure 2, as K increases, the stable positive steady-state loses its stability through a Hopf bifurcation point and gives rise to a limit cycle. Further increasing K collapses the cyclic behaviour through a homoclinic bifurcation and returns the dynamics to a stable steady-state

10 294 H. Wang et al. a 1.8 b : red Mechanistic model: black.8 : red Mechanistic model: black producer Light intensity represented by K, mg C/L consumer Light intensity represented by K, mg C/L Figure 2. Bifurcation diagrams for Model 3 with respect to K. a producer vs. K; b consumer vs. K. behavior. The consumer goes etinct for etremely small K because of low food quantity, or for etremely large K because of the low food quality panel b in Figure 2. Solutions of Model 3 are compared to that of the for various K values in Figure 3. For small or large K steady-state cases, panels a and d in Figure 3, our model and the have almost identical solutions. For the intermediate K values panels b and c in Figure 3, they are slightly different quantitatively. However, when K is near the homoclinic bifurcation point see Figure 4, they are completely different. Near this point, the solution sensitivity with respect to K is very high. Nevertheless, in general, we can state that see the bifurcation diagrams Figure 2 the LKE model is a very good approimation of the mechanistic Model 3. a.4 species densities mechanistic model K=.25 mg C/L producer consumer b.7 K=.75 mg C/L.6 mechanistic model species densities producer consumer time day.1 mechanistic model time day c mechanistic model producer consumer d K=2 mg C/L producer consumer species densities species densities mechanistic model.35.3 K=1 mg C/L mechanistic model time day time day Figure 3. Comparison of solutions of the mechanistic model and the with different carrying capacities. a coeistence at a steady state; b coeistence with oscillations; c coeistence at a steady state with a higher producer/grazer ratio; d etinction of the grazer..5

11 Journal of Biological Dynamics 295 species densities producer consumer mechanistic model.1 K=.95 mg C/L time day Figure 4. Comparison of Model 3 with the with the following parameter values: r =.93, c =.75, ĉ =.2, a =.25, â =.8, e =.74, d =.22, ˆd =.5, =.4, q =.4, T = Discussion We constructed and analyzed a stoichiometric predator-prey model that eplicitly tracks the quantity and the nutritional quality of the prey. We achieve this by mechanistically accounting for the content of two essential elements in this simple food web, C and P. Carbon is considered to be energy quantity equivalent, while the concentration of P or P:C ratio reflects nutritional quality of the prey. This model builds upon the by epanding it from a two-dimensional system to a four-dimensional one. The additional two equations track P in the prey and in the media that supports the prey. The assumes that the prey is etremely efficient at uptaking nutrient P from the media; hence, Assumption A3, no P in the media. Interestingly, if we assume in Model 4 the infinite uptake efficiency of prey α, then our model converges naturally to the. Here is the argument. Observe that the nutrient p in the producer and P in the media possess a dynamics that is much faster than growth dynamics of producer and consumer. We claim that Model 4 has almost the same dynamics as that of the. This can be seen by a simple application of the quasi-steady-state argument on the nutrient equation 4c. This yields p as a function of and y, which is αt y/α + βy + d. Thus, Equation 4 can be reduced to d dy = r 1 =ê min 1, mink, αt y/α + βy + d/q} [αt y/α + βy + d]/ } βy ˆdy. βy, The plant phosphorus in the is T y, a function of y, following Assumption A3. The phosphorus outside the grazer, T y, is distributed between the plant and the media, and the plant phosphorus has the fraction α/α + βy + d. Therefore, the plant phosphorus should be a function of both and y. α is large in fast nutrient process, and α/α + βy + d 1 for sufficiently large α; then, the plant phosphorus is almost T y for fast nutrient process. Hence, the can be regarded as the limiting case of Model 4 when α. However, even with finite nutrient uptake rates for the producer, our model behaves both qualitatively and quantitatively similar to the as bifurcation diagrams 2 and simulations 3 show. The eception is the interval for K near the point where the predator limitation switches from C to P from food quantity to food quality, where there is a quantitative difference between the two models. This raises the question whether the significant similarity between the two models justifies the additional compleity of our model. We certainly think so for the following reasons. 6a 6b

12 296 H. Wang et al. In applications, such as the control of eutrophication in lakes, it is the actual concentration of P in the water that is of crucial importance, and our model eplicitly tracks it. This mechanistical model formulation also provides an additional significant benefit; our model can be easily epanded to multiple producers and grazers while maintaining its structure. This later property is not shared by the because Assumption A3 does not leave a mechanistic way to distribute P among multiple producers. Acknowledgements The research of Yang Kuang and Hao Wang is supported in part by DMS and DMS/NIGMS Correspondence should be directed to Hao Wang. References [1] T. Andersen, Pelagic Nutrient Cycles: Herbivores as Sourced and Sinks for Nutrients. Springer-Verlag, Berlin, [2] T. Andersen, J.J. Elser, and D.O. Hessen, Stoichiometry and population dynamics, Ecol. Lett pp [3] J.J. Elser, et al., Nutritional constraints in terrestrial and freshwater food webs, Nature 48 2 pp [4] J.P. Grover, Stoichiometry, herbivory and competition for nutrients: Simple models based on planktonic ecosystems, J. Theor. Biol pp [5] D.O. Hessen and B. Bjerkeng, A model approach to planktonic stoichiometry and consumercresource stability, Freshwater Biol pp [6] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol pp [7] Y. Kuang, J. Huisman, and J.J. Elser, Stoichiometric plant-herbivore models and their interpretation, Math. Biosc. Eng pp [8] I. Loladze, Y. Kuang, and J.J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math. Biol pp [9] I. Loladze, et al., Competition and stoichiometry: Coeistence of two predators on one prey, Theo. Popu. Biol pp [1] E.B. Muller, et al. Stoichiometric food quality and herbivore dynamics, Ecol. Lett pp [11] R.W. Sterner and J.J. Elser, Ecological Stoichiometry, Princeton University, Princeton, NJ, 22.