Stoichiometry in population dynamics and its implications

Size: px

Start display at page:

Download "Stoichiometry in population dynamics and its implications"

Asher McDonald
6 years ago
Views:

1 Stoichiometry in population dynamics and its implications Yang Kuang Department of Mathematics and Statistics Arizona State University ICMB, 7. Wuyishan, China p. 1/48

2 Main Collaborators Irakli Loladze(Univ. Nebraska, Lincoln), Jim Elser(Biology, ASU), William F. Fagan (Biology, Univ. Maryland), Jef Huisman (Inst. for Biodiversity & Ecosystem Dynamics, Amsterdam), Hao Wang, ASU Most papers can be downloaded at ICMB, 7. Wuyishan, China p. 2/48

3 Ecological Stoichiometry Stoichiometry is the accounting, or math, behind chemistry. It deals with the balance of multiple chemical elements in chemical reactions. Ecological stoichiometry is the study of the balance of energy and multiple chemical resources (elements) in ecological interactions. ICMB, 7. Wuyishan, China p. 3/48

GRH T he G rowth R ate Hypothes is natural s elec tion on growth rate c ellular inves tment (r ibos ome c ontent) bioc hemic al inves tment (R NA: pr otein) B ody C : N: P food quality constraints on

4 GRH T he G rowth R ate Hypothes is natural s elec tion on growth rate c ellular inves tment (r ibos ome c ontent) bioc hemic al inves tment (R NA: pr otein) B ody C : N: P food quality constraints on growth / reproduction resource competition trophic efficiency nutrient recycling T he first picture of a ribosome. C ate et al. (1999) S cience 285: B ased on: E lser, J.J., D.R. Dobberfuhl, N.A. MacK ay, and J.H. S champel. Organism size, life history, and N:P stoichiometry: toward a unified view of cellular and ecosystem processes. BioS cience 46: ICMB, 7. Wuyishan, China p. 4/48

leaf biomass with less nitrogen content.

and 49% in the 1 st and 2 nd generations

5 Leaf mites Elevated CO 2 and Leaf Mites Study: Joutei, Roy, Van Impe and Le brun published in 2. Elevating the level of CO 2 caused larger leaf biomass with less nitrogen content. The leaf mites progeny were reduced by 34% and 49% in the 1 st and 2 nd generations and later stages of development were reduced. ICMB, 7. Wuyishan, China p. 5/48

6 Daphnia (C X N Y P Z ) inorganic + (C X N Y P Z ) autotroph + light -> Q (C X N Y P Z )' autotroph + (C X N Y P Z ) inorganic (C X N Y P Z ) prey + (C X N Y P Z ) predator -> Q (C X N Y P Z ) predator + (C X N Y P Z ) waste F rom: E lser, J.J., and J. Urabe T he stoichiometry of consumer-driven nutrient recycling: theory, observations, and consequences. E cology 8: ICMB, 7. Wuyishan, China p. 6/48

7 Introduction Mathematical biologists have built on variants of the Lotka Volterra equations and in almost all cases have adopted the pure physical science s single-currency (energy) approach to understanding population dynamics. However, biomass production requires more than just energy. It is crucially dependent on the chemical compositions of both the consumer species and food resources. In this talk, we explore how depicting organisms as built of more than one thing (for example, C and an important nutrient, such as P) in stoichiometrically explicit models results in qualitatively different and realistic predictions about the resulting dynamics. ICMB, 7. Wuyishan, China p. 7/48

8 LKE Model Model of Loladze, Kuang and Elser (LKE model), appeared in Bull. Math Biol., 2. [ dx dt = bx 1 dy dt = c min ] x f(x)y, min(k, (P) θy)/q) f(x)y dy. ( 1, P θy θx (1) ICMB, 7. Wuyishan, China p. 8/48

9 LKE Model Assumptions A1. The total mass of phosphorus P in the entire system is fixed, i.e. the system is closed for phosphorus with total of P (mg P/l). A2. Phosphorus to carbon ratio (P:C) in the producer varies, but it never falls below a minimum q (mg P/mg C); the grazer maintains a constant P:C ratio, denoted byθ (mg P/mg C). A3. All phosphorus in the system is divided into two pools: phosphorus in the grazer and phosphorus in the producer. ICMB, 7. Wuyishan, China p. 9/48

10 LKE model dynamics T heoretic al T es t of L ight: Nutrient E ffec ts Model of Loladze, K uang and E ls er (modified from model of T. Anders en) x' (t) = bx (1 - x min[k, (P - θy) / q] ) - f (x)y G razer y' (t) = εmin (1, (P - θy) / x ) f (x)y - dy θ P roduc er F rom: Loladze, I, Y. K uang, and J.J. E lser. 2. S toichiometry in producer-grazer systems: linking energy flow and element cycling. B ull. Math. B iol. 62: ICMB, 7. Wuyishan, China p. 1/48

11 LKE model dynamics T heoretic al T es t of L ight: Nutrient E ffec ts Model of Loladze, K uang and E ls er (modified from model of T. Anders en) light light light F rom: Loladze, I, Y. K uang, and J.J. E lser. 2. S toichiometry in producer-grazer systems: linking energy flow and element cycling. B ull. Math. B iol. 62: ICMB, 7. Wuyishan, China p. 11/48

12 LKE model dynamics T heoretic al T es t of L ight: Nutrient E ffec ts Model of Loladze, K uang and E ls er (modified from model of T. Anders en) G razer light F rom: Loladze, I, Y. K uang, and J.J. E lser. 2. S toichiometry in producer-grazer systems: linking energy flow and element cycling. B ull. Math. B iol. 62: ICMB, 7. Wuyishan, China p. 12/48

13 Experimental results Aquatron Dynamic s C onsumer biomass (mgc l -1 ) Food biomass (mg C l -1 ) A L ow L ight (4 µe / sq m / s) Algal P : C Daphnia High L ight (31 µe / sq m / s ) B (E xtra) High L ight (38 µe / sq m / s ) C P :C ratio of food (x 1-3 ) Algal C Days C trans fer efficienc y: ~3% C trans fer efficienc y: ~7% Urabe, J., J.J. E lser, M. K yle, T. S ekino and Z. K awabata. 22. Herbivorous animals can mitigate unfavorable ratios of energy and material supplies by enhancing nutrient recycling. E cology Letters: in press. ICMB, 7. Wuyishan, China p. 13/48

14 Experimental results Aquatron Dynamic s 1 L ow L ight (4 µe / sq m / s) A B C D. pulic aria High L ight (31 µe / sq m / s ) (E xtra) High L ight (38 µe / sq m / s ) Daphnia (indiv. l -1 ) 1 1 D. magna Days exc lus ion c oexis tence Urabe, J., J.J. E lser, M. K yle, T. S ekino and Z. K awabata. 22. Herbivorous animals can mitigate unfavorable ratios of energy and material supplies by enhancing nutrient recycling. E cology Letters: in press. ICMB, 7. Wuyishan, China p. 14/48

15 Take home messages Stoichiometric models incorporate both food quantity and food quality effects in a single framework, appear to stabilize predator prey systems while simultaneously producing rich dynamics with alternative domains of attraction and occasionally counterintuitive outcomes, such as coexistence of more than one predator species on a single-prey item and decreased herbivore performance in response to increased plant growth rate. Stoichiometric theory has tremendous potential for both quantitative and qualitative improvements in the predictive power of mathematical population models in the study of both ecological and evolutional dynamics. ICMB, 7. Wuyishan, China p. 15/48

16 Mechanistic formulation If we let P p,p z and P f be the phosphorous in autotroph, phosphorous in herbivore, and the free phosphorous respectively, then P t = P p + P z + P f. Let x = x(t) be the autotroph density, y = y(t) be the herbivore density and Q = Q(t) be the autotroph s cell quota for P, then P p = Qx and P z = θy. Hence P t = P f + Qx + θy. (2) In the following, we let q be the autotroph s minimal cell quota for P, µ m be the autotroph s true maximal growth rate, D be its death rate and f(x) be the herbivore s ingestion rate (functional response). ICMB, 7. Wuyishan, China p. 16/48

17 Mechanistic formulation By Droop equation, we have the following equation for the autotroph growth ( dx dt = µ m 1 q ) x Dx f(x)y. (3) Q ICMB, 7. Wuyishan, China p. 17/48

18 Mechanistic formulation Let e be the herbivore s yield constant which measures the conversion rate of ingested autotroph into its own biomass when the autotroph is P rich (when Q θ) and d be the specific loss rate of herbivore that includes metabolic losses and death. If the autotroph is P poor (when Q < θ), then the conversion rate suffers a reduction and it becomes eq/θ. This approach follows the Liebig s (184) minimum principle and is used in Loladze et al. s (2) model formulation. We have the following growth equation for herbivore dy dt = e min ( 1, Q θ ) f(x)y dy. (4) ICMB, 7. Wuyishan, China p. 18/48

19 Mechanistic formulation Finally, we need an equation governing the dynamics of Q, the autotroph s cell quota for P. We assume that Q s recruitment comes proportionally from the free phosphorous (αp f ) and its depletion due to the cell growth is µ m (Q q). This results in the following simple equation dq dt = αp f µ m (Q q). (5) Since Q() q, mathematically, it is easy to see that this ensures that Q(t) q for all t >. ICMB, 7. Wuyishan, China p. 19/48

20 Mechanistic formulation Since the cell metabolic process operates in a much fast pace than the growth of total biomass of either species, we approximate Q(t) by the solution of αp f µ m (Q q) =. (6) We have the following autotroph-herbivore model when P is the only limiting element: dx dt dy dt [ = (µ m D)x 1 = e min ( 1, Q θ x + µ m α 1 [(µ m D)/µ m ][µ m α 1 + (P t θy)/q] ) f(x)y dy. ] f(x)y. (7) ICMB, 7. Wuyishan, China p. 2/48

21 Mechanistic formulation If only C is limiting the autotroph s growth, then the traditional autotroph equation can be used. ( dx dt = bx 1 x ) f(x)y (8) K where b = µ m D is the net autotroph growth rate. Applying Leipig s minimum principle, we obtain the following autotroph-herbivore model with two limiting elements dx dt dy dt [ ( x = bx 1 max = emin ( 1, Q θ K, ) f(x)y dy. x + µ m α 1 [(µ m D)/µ m ][µ m α 1 + (P t θy)/q] )] f(x)y. (9) ICMB, 7. Wuyishan, China p. 21/48

22 Mechanistic formulation If we, in addition, assume that the natural autotroph death rate D is far less than its true maximal growth rate, then we can approximate the value of (µ m D)/µ m by 1. Together with the assumption α tends to, the above model becomes dx dt dy dt [ ( x = bx 1 max = e min ( 1, Q θ K, ) f(x)y dy. )] x (P t θy)/q f(x)y. (1) ICMB, 7. Wuyishan, China p. 22/48

23 Mechanistic formulation As α tends to, we see that Q tends to (P t θy)/x. The above model s exactly the same as the LKE (Loladze, Kuang and Elser: Stoichiometry in producer-grazer systems: linking energy flow and element cycling, Bull. Math. Biol., 62, (2)) model: dx dt dy dt ( = bx = e min ) x f(x)y, ( min(k, (P t θy)/q) 1, (P ) t θy)/x f(x)y dy. θ 1 (11) ICMB, 7. Wuyishan, China p. 23/48

24 SIMULATION.7.6 A: Low light I in =5 plant isocline herbivore isocline.7.6 B: Medium light I in =1 plant isocline herbivore isocline.5.5 herbivore herbivore High light I in =2 plant isocline herbivore isocline.6 Very high light I in =35 plant isocline herbivore isocline.5.5 herbivore herbivore plant plant ICMB, 7. Wuyishan, China p. 24/48

25 BIFURCATION diagram Bifurcation diagram for a stoichiometric plant herbivore model plant min plant plant max min herbivore plant max min herbivore herbivore max min quality herbivore indicator max quality indicator 1.4 plant, herbivore Iin, the incident light intensity ICMB, 7. Wuyishan, China p. 25/48

26 Single resource with two consumers Let us start with a conventional model, which describes a system of two consumers feeding on one biotic resource. dx dt dy 1 dt dy 2 dt = rx ( 1 x ) K = e 1 f 1 (x)y 1 d 1 y 1 = e 2 f 2 (x)y 2 d 2 y 2 f 1 (x)y 1 f 2 (x)y 2 (12) ICMB, 7. Wuyishan, China p. 26/48

27 Single resource with two consumers dx dt ( ) x = rx 1 min(k,(p s 1 y 1 s 2 y 2 ) /q) = F(x, y 1, y 2 ) f 1 (x)y 1 f 2 (x)y 2 dy 1 dt dy 2 dt = e 1 min = e 2 min ( 1, (P s ) 1y 1 s 2 y 2 ) f 1 (x)y 1 d 1 y 1 = G 1 (x, y 1, y 2 ) xs 1 ( 1, (P s ) 1y 1 s 2 y 2 ) f 2 (x)y 2 d 2 y 2 = G 2 (x, y 1, y 2 ) xs 2 (13) ICMB, 7. Wuyishan, China p. 27/48

28 Simulation Bifurcation diagram for a stoichiometric one plant two herbivores model plant min plant max herbivore y min herbivore y max herbivore z min herbivore z max quality indicator for y quality indicator for z 1.2 plant, herbivores K, the carrying capacity ICMB, 7. Wuyishan, China p. 28/48

29 Simulation The coexistence of all species at a stable positive equilibrium is possible. (r = 1.4,c 1 =.63,c 2 =.6,a 1 =.45,a 2 =.36,e 1 =.85,e 2 =.8,P =.36,q =.3,s 1 =.38,s 2 =.25,d 1 =.1,d 2 =.12. plant Bifurcation diagram for a stoichiometric one plant two herbivores model plant min plant plant max min plant max herbivores y herbivore y min herbivore y max y min quality herbivore indicator y max for y quality indicator for y herbivores z herbivore z min herbivore z max z min quality herbivore indicator z max for z quality indicator for z K, the carrying capacity ICMB, 7. Wuyishan, China p. 29/48

30 An alternative formulation Let x be the density of carbon content in the producer, p be the density of phosphorus content in the producer, y is the density of carbon contents in the grazer, P is the density of free phosphorus in media. The following alternative stoichiometric model is more mechanistical. ( dx dt = rx 1 dy dt = ê min ) x f(x)y, min{k, p/q} } f(x)y ˆdy, { 1, p/x θ (14a) (14b) dp dt = g(p)x p f(x)y dp, x (14c) dp dt = g(p)x + dp + θ ˆdy ( p { + x ê min θ, p }) f(x)y (14d) x ICMB, 7. Wuyishan, China p. 3/48

31 Comparison species densities LKE model mechanistic model K=.25 (mg C)/L producer consumer species densities K=.75 (mg C)/L mechanistic model producer consumer LKE model.1 LKE model time (day) (a) A coex. steady state.1 LKE model mechanistic model time (day) (b) Oscillation mechanistic model producer consumer K=2 (mg C)/L producer consumer species densities LKE model species densities LKE model mechanistic model.35.3 K=1 (mg C)/L mechanistic model time (day) (c) Another coex. equil time (day) (d) extinction of the grazer ICMB, 7. Wuyishan, China p. 31/48

32 Multiple consumer species, nutrients (new l consumers-m producers model with n nutrients is dx j dt { j = r j x j min 1 x } j K,1 x j f i,j (x j )y i, min s {p j,s /q j,s } i { { j 1,min (p }} j,s/x j )f i,j (x j ) s j f f i,j (x j )y i d i y i, i,j(x j )θ i,s dy i dt = e i min dp j,s dt = g j,s (T s j p j,s i θ i,s y i )x j p j,s x j i j f i,j (x j )y i d j,s p j,s (15a) (15b) (15c) i = 1,2,..., l, j = 1,2,..., m, s = 1,2,..., n. Potentially, at most m (n + 1) consumers can coexist in steady-state sense. ICMB, 7. Wuyishan, China p. 32/48

33 Multiple consumer species, nutrients x = y i = x bx 1 ( ) ( ) k k min(k, N 1 s 1i y i /q 1,..., N n s ni y i /q n ) i=1 i=1 m f i (x)y i i=1 ( ) ( ) k k N 1 s 1i y i N n s ni y i e i=1 i=1 i min 1,,..., f xs 1i xs ni i (x)y i d i y i (16) N i is the total amount of i-th nutrient in the system, q i is the resource s minimal i-th nutrient content, s ij is the j-th consumer s constant (homeostatic) i-th nutrient content. ICMB, 7. Wuyishan, China p. 33/48

34 Simulation densities a: Coexistence of three consumers on one resources (with stoichiometry) x y 1 y 2 y 3 densities t (days) b: Coexistence of two consumers on one resources (without stoichiometry) x y 1 y 2 y t (days) ICMB, 7. Wuyishan, China p. 34/48

35 Simulation.8.7 x y 1 y 2 y t Three consumers coexisting (in an oscillatory way) on a single resource (with C, P and N). ICMB, 7. Wuyishan, China p. 35/48

36 Simulation Bifurcation diagram for a stoichiometric one plant two herbivores model plant, herbivores plant min plant max herbivore y 1 min herbivore y 1 max herbivore y 2 min herbivore y 2 max herbivore y 3 min herbivore y 3 max quality indicator for y 1 quality indicator for y 2 quality indicator for y K, the carrying capacity ICMB, 7. Wuyishan, China p. 36/48

37 Simulation 1 a: Coexistence of four consumers on two resources (with stoichiometry) densities x 1 y 1 y 2 x 2 y 5 y t (days) 1 b: Coexistence of only two consumers on two resources (without stoichiometry) densities x 1 y 1 y 2 x 2 y 5 y t (days) ICMB, 7. Wuyishan, China p. 37/48

38 Coexistence Our experience suggests that the ultimate dynamics of model (16) will depend on both model parameters and initial population densities. We can speculate that if one biotic resource provides n essential chemical elements for competing consumers, it may support up to n different competitors at a stable equilibrium. ICMB, 7. Wuyishan, China p. 38/48

39 Coexistence The formulation of a plausible and tractable mathematical version of model (16) with more than one biotic resource is challenging. This difficulty stems from the competition among resource species, which forces one either to consider free pools of essential elements (i.e., not bound in any biomass) and model their uptake, or to find some other way to distribute each essential element among all resource species. A mathematically easier case can be imagined in a patchy habitat, where specialist consumers in each patch exploit a distinct biotic resource that provides n essential chemical elements. Then, m distinct biotic resources can support up to m n consumer species at a stable equilibrium. ICMB, 7. Wuyishan, China p. 39/48

40 A Discrete LKE Model: Our Model We assume that the per capita growth rates in (1) change only at the time of each measurement. Incorporating this aspect in (1) yields the following modified system 1 x(t) dx(t) dt = b [ 1 x([t]) min(k,(p θy([t]))/q) ] f(x([t]))y([t]), x([t]) 1 y(t) dy(t) dt = cmin ( 1, P θy([t]) ) f(x([t])) d, t,1, 2,. θx([t]) (17) Here [t] denotes the integer part of t (, + ). ICMB, 7. Wuyishan, China p. 4/48

41 A Discrete LKE Model: Our Model On any interval of the form [n,n + 1),k =, 1, 2, we can integrate (17) and obtain for n t < n + 1,n =, 1, 2, x(t) = x(n) exp y(t) = y(n)exp {[ b bx(n) min(k,(p θy(n))/q) f(x(n))y(n) ] } (t n) x(n) {[ ( c min 1, P θy(n) ) ] } f(x(n)) d (t n). θx(n) (18) ICMB, 7. Wuyishan, China p. 41/48

42 A Discrete LKE Model: Our Model Letting t n + 1, we obtain from (18) that x(n + 1) = x(n) exp y(n + 1) = y(n) exp { b bx(n) min(k,(p θy(n))/q) f(x(n))y(n) }, x(n) { ( c min 1, P θy(n) ) } f(x(n)) d. θx(n) (19) This is a discrete time analogue of LKE model (1). In the following sections, we will focus our attention on system (19). We assume that x() > and P/θ > y() >. ICMB, 7. Wuyishan, China p. 42/48

43 Simulation In the following, we assume that p(x) = r/(a + x). x(n + 1) = x(n) exp y(n + 1) = y(n)exp { b bx(n) min(k,(p θy(n))/q) cy(n) a + x(n) { ( emin 1, P θy(n) ) } cx(n) θx(n) a + x(n) d, } (2) and [ dx dt = bx 1 x min(k,(p θy)/q) ] cxy a + x ( dy dt = emin 1, P θy ) cxy θx a + x dy (21). ICMB, 7. Wuyishan, China p. 43/48

44 Simulation Parameter Value Unit P Total phosphorus.25 mg P l 1 e Maximal production efficiency in carbon terms.8 b Maximal growth rate of the producer 1.2 day 1 d Grazer loss rate (includes respiration).25 day 1 θ Grazer constant P/C.3 (mg P)/(mg C) q Producer minimal P/C.38 (mg P)/(mg C) c Maximum ingestion rate of the grazer.81 day 1 a Half-saturation of grazer ingestion response.25 mg C l 1 K Producer carrying capacity limited by light mg C l 1 ICMB, 7. Wuyishan, China p. 44/48

45 Simulation.5.4 (d 1 ) K=.25 producer grazer.5.4 (c 1 ) K=.25 producer grazer time (d 2 ) K=.75 producer grazer time time (c 2 ) K=.75 producer grazer time 1.8 (d ) K=1. 3 producer grazer grazer 1.8 (c ) K=1. 3 producer grazer grazer time (d 4 ) K= time (c 4 ) K= producer grazer producer grazer producer grazer time time ICMB, 7. Wuyishan, China p. 45/48

46 Simulation.7 (a) bifurcation curve for (5.1).7 (b) bifurcation curve for (5.2) grazer grazer K K ICMB, 7. Wuyishan, China p. 46/48

47 Chaos can drive predators to extinction (a) Producer bifurcation curve for (5.1): K=1.6 8 (b) Grazer bifurcation curve for (5.1): K= producer 4 grazer b b (c) Producer bifurcation curve for (5.2): K=1.6 2 (d) Grazer bifurcation curve for (5.2): K= producer 1 grazer b b ICMB, 7. Wuyishan, China p. 47/48

48 Discussion The continuous and the discrete stoichiometric models exhibit same qualitative phenomena. This suggests the robustness of underlying effects of prey quality on predator-prey dynamics. Moreover, the discrete model shows that chaos can arise in stoichiometric system due to variation in food quality. The simple discrete model clearly shows period-doubling route to chaos, which happens within biologically plausible parameter range. This supports Andersen s (1997) argument on the possibility of chaos in stoichiometric systems. ICMB, 7. Wuyishan, China p. 48/48

A Producer-Consumer Model With Stoichiometry

A Producer-Consumer Model With Stoichiometry Plan B project toward the completion of the Master of Science degree in Mathematics at University of Minnesota Duluth Respectfully submitted by Laura Joan Zimmermann