Luận án tiến sĩ: Mathematical analysis of trophic interactions: From bacteria competition to lemming cycles

MATHEMATICAL ANALYSIS OF TROPHIC INTERACTIONS: FROM BACTERIA COMPETITION TO LEMMING CYCLES by Hao Wang A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy ARIZONA STATE UNIVERSITY May 2007 UMI Number: 3243897 Copyright 2007 by Wang, Hao All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted.

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ProQuest Information and Learning Company 300 North Zeeb Road P. Box 1346 Ann Arbor, MI 48106-1346 MATHEMATICAL ANALYSIS OF TROPHIC INTERACTIONS: FROM BACTERIA COMPETITION TO LEMMING CYCLES by Hao Wang has been approved November 2006 NS woe =“ we) Supervisory Committee ACCEPTED: B o w fhe Chait Bean, he of Ll Studies ABSTRACT Mechanistic and phenomenological models and careful parameter estima- tions are presented through both aquatic and terrestrial ecosystems. The sto- ichiometric modeling of the bacteria-algae lake system is relatively new, while the lemming population cycle has attracted the attention of several generations of theoretical and experimental biologists and continues to be an issue of controversy. Bacteria-algae interaction in epilimnion is modeled with explicit consider- ation of carbon (energy) and phosphorus (nutrient).

Global qualitative analysis and bifurcation diagrams of this model are presented. Competition of bacterial strains are modeled to examine Nishimura’s hypothesis that in severely P-limited environments, such as Lake Biwa, P limitation exerts more severe constraints on the growth of bacterial groups with higher nucleic acid contents, which allows low nucleic acid bacteria to be competitive. Through a series of carefully derived models of the well documented high- amplitude, large-period fluctuations of lemming populations at Point Barrow, one can argue that, when appropriately formulated, autonomous differential equations may capture much of the desirable rich dynamics such as the existence of a periodic solution with period and amplitude close to that of approximately periodic solu- tions produced by the more natural but mathematically daunting nonautonomous models. This, together with the bifurcation analysis, indicates that neither sea- sonal factors, nor the moss growth rate and lemming death rate, are the main determinants of the observed multi-year lemming cycles.

What ecological factors control population cycles? For some species — ili collared lemmings and snowshoe hares in particular—maturation delay of preda- tors and the functional response of predation appear to be the primary determi- nants. Maturation delay almost completely determines the cycle period, whereas the functional response greatly affects its amplitude and even its existence. This result is obtained from sensitivity analysis of all the parameters and comparison of the lemming-stoat and hare-lynx systems. iv To my wife ACKNOWLEDGMENTS First of all I would like to thank my advisor Dr.

Yang Kuang for offering me precious chances to implement experiments and work with biologists, also his encouragement during my Ph.D study at Arizona State University. I would like to thank my co-advisor Dr. Hal Smith for his numerous discussions on mathematical analysis and invaluable help on all aspects, and Dr. John Nagy for their many insightful discussions in biology.

I also thank Dr. Olivier Gilg, Dr. Sebastian Diehl, Dr. James Grover, Dr.

Val Smith, Dr. Irakli Loladze and Dr. Jiaxu Li for helpful discussions and suggestions. In addition, thanks to Dr.

Horst Thieme and Dr. Sharon Crook for their interesting courses. I am blessed to work with these excellent mathematicians and biologists. Last but not the least, I am forever indebted to my wife, Qiangying Cao, for her pure love and inspiration.

Hao Wang December 25, 2006 Arizona State University vi TABLE OF CONTENTS Page LIST OF TABLES. 0000200040, ix LIST OF FIGURES. eee, CHAPTER 1 BIOLOGICAL BACKGROUND. Organization of Dissertation.

CHAPTER 2 STOICHIOMETRY OF LAKE BACTERIA AND ALGAE 1. ch ho Algae Dynamics. Persistence and Invasion of Bacteria. 4, Competing Bacterial Strains.

CHAPTER 3 FOOD-LEMMING SYSTEMS IN ALASKA 1. ch vii Page CHAPTER 4 LEMMING-PREDATOR SYSTEMS IN GREENLAND. Simple Lemming-Stoat Model. Numerical Solution and Limitation.

Lemming-Stoat Delay Model. Empirical Data Fitting. Snowshoe Hare-Lynx Delay Model. Comparison and Interpretation.

Summary and Discussion.0 0 05 eee 89 CHAPTER 5 CONCLUDING REMARKS. Key Points of Thesis. kg kg Ta 93 REFERENCES. Quàva 98 APPENDIX A MATLAB PROGRAMS.

co 108 vill LIST OF TABLES Table Page Variables in bacteria-algae system (2. 16 Parameters in bacteria-algae system (2. 17 Parameters in Barrow model (3. 47 Parameters in moss-lemming models.

53 Comparison of all four lemming models. 65 Parameters in lemming-stoat systems. 70 Empirical lemming data from Olivier Gilg ., 76 Parameters in hare-lynx system (4. 84 ix LIST OF FIGURES Figure A flow chart for GRH (Elser et al.- The empirical field data for the 4-year lemming cycle.

This is mod- ified from Gilg et al. ees The structure of this thesis. ch 9 A checklist of all the locations and species analyzed in the thesis. 9 The logical relationship between algae and bacteria in a lake system.

12 The cartoon lake system for our mathematical models. 14 Graph of algae system (2.2) to check that system is competitive. This is observed from the Jacobian matrix of system (2.2) in the proof of Theorem 3. ho 25 A bifurcation diagram for system (2.2), illustrating that the shal- lower the mixed layer the better for algae in system (2.

This bifurcation diagram confirms our mathematical findings. When Ry > 1, the algae extinction equilibrium is unstable, and the only positive equilibrium appears to be globally attractive. The branch- ing point occurs at Ro = 1. When Rp < 1, there is no positive equilibrium and the algae extinction equilibrium is globally attract- ing.

This numerical result is generated by the continuation software ‘MatCont’ in MATLAB. Q Q Q he Figure Page Phase plane when Ry > 1 for system (2. The algae extinction equilibrium Ep = (0,Q, Pn) is globally attracting on the subspace 9 = {2 € | A = 0} but a uniform weak repeller for Q, = {x € Q| AO}, and A is persistent in this case. Algae dynamics without bacteria (system (2.2)) with respect to different depths of the mixed layer: All the variables approach the positive equilibrium in about two months.

The first two figures suggest that with deeper mixing depths, P increases, but A de- creases. On the other hand, average light intensity should decrease with larger depths, because of shading effect. Hence, algal growth is more limited by energy than nutrient in a lake with a deeper mixed layer. This numerical simulation is generated by ‘ode23s’ in MATLAB using the initial conditions: A = 20,P = 0.

27 xi Figure Page 11. An abstract phase plane diagram for system (2.1) when Ro > 1 and R, > 1. Q,P are placed on one axis (say x-axis), A,C are placed on another axis (say y-axis) and B is on the vertical axis (z-axis). Extinction equilibrium eo = (0,Q, Pin, 0,0) is globally attracting on the subspace {x € 9 | A = B = C =0}, but a repeller for Q, = {ce € 2 | B = 0).

Bacteria extinction only equilibrium e, = (A,Q, P,0,C) is globally attracting on the subspace M2, but a repeller for Q) = {x €2| BO}. B is persistent, and at least one coexistence equilibrium exists. eee ee es 34 12. Regions of P;, versus In, for survival and extinction of bacteria and algae.

Both algae and bacteria go extinct (Ry < 1) in the grey region. Both algae and bacteria survive (Ro > 1,R1 > 1) in the white region. Algae survive but bacteria go extinct (Ro > 1, Ri < 1) in the red/dark region. We run simulations of system (2.1) for each pair of (Jin, Pin), plot the point in grey if both A and B go to zero, in white if both persist, and in red/dark if A persists but B GOES tO ZETO.

kh kh ky Q. A quantitative relationship between bacteria and algae at the low P level (Pi. = 30), illustrating that B: A ratio is decreasing in solar energy input. They also confirm the qualitative result (Figure 12).

These two figures are generated by ‘MatCont’ in MATLAB. 37 xil Figure Page 14. We examine Nishimura’s hypotheses using system (2. These two figures are generated by ‘MatCont’ in MATLAB.

Under different lake environments, different bacterial strains dom- Ko = 0.1, with units in Table 2. These simulations are generated by ‘ode23s’ in MATLAB with the initial conditions: A = 20,Q = 0. When the mixed layer is shallow with z„ = 1, system (2.1) exhibits complex dynamics. These simulations are generated by ‘ode23s’ in MATLAB with the initial conditions: A = 350,Q@ = 0.

Bifurcation diagrams of system (2.1) for depth of epilimnion. These are generated by 'MatCont'in MATLAB. Numerical simulation of Barrow model (3.1) with the median val- ues of parameters shown in Table 3. This numerical simulation is generated by ‘ode23s’ in MATLAB with the initial conditions: V=100,M@=1000,H=20.

Numerical simulation of nonautonomous moss-lemming model (3.2) with the median values of parameters shown in Table 3 and Z = 0. This numerical simulation is generated by ‘ode23s’ in MAT- LAB with the initial condition: M = 1000, H = 20. xi Figure Page 20. u(t) and d(t) are rep- resented by the red continuous curves.

Using mean values of Us, ds,dy, we compare u(t),d(t) with u(r), d(r) statistically. The standard/average error of u(t) with respect to u(r(t)) is 1.538 and the relative error is 1. The standard/average er- ror of d(t) with respect to d(r(t)) is 0.401 and the relative error is 0. These two continuous functions are constructed manually.

Then we plot each of them with the cor- responding discontinuous function in one figure and also calculate CITOTS. A typical solution of the nonautonomous moss-lemming model (3.4) with the median values of parameters shown in Table 3 and / = 0. This numerical simulation is generated by ‘ode23s’ in MATLAB with the initial conditions: z = 1000,=20. A typical solution of the autonomous moss-lemming model (3.5) with the median values of parameters shown in Table 3 and / = 0.

This numerical simulation is generated by ‘ode23s’ in MATLAB with the initial conditions: x = 1000,y=20. XIV Figure Page 23. Comparison of solutions of nonautonomous model (3.4) and au- tonomous model (3. Numerical solutions of both models are posed in the same figure with the same initial conditions: © = 1000,y= 20.

Bifurcation diagrams and limit cycles for the autonomous system (3. These bifurcation diagrams are generated by plotting maxi- mums and minimums of all the eventually stabilized oscillations. Period diagrams for the autonomous system (3. These period bi- furcation diagrams are generated by modifying the ‘ode23’ program in MATLAB with Jiaxu Li’s great help.

The location in NE Greenland where field experiments were carried out by Olivier Gilg and colleagues. Stoat functional response to lemmings (Gilg et al. A typical solution of the simple lemming-stoat system (4.1) with parameter values in Table 6. This numerical solution is generated by ‘ode23s’ in MATLAB with the initial conditions: z = 0.

We use the command ‘plotyy’ to plot this multi-scale graph. 72 xv Figure Page 29. Lemming-stoat dynamics predicted by the delay system (4.3) com- pared to empirical data (points). The empirical data is in Ta- ble 7.

Stoat density is collected in winters; hence, stoat data points are shifted to the left half a year. The numerical predic- tions are generated by ‘dde23’ in MATLAB with the initial condi- tions: x = 0.