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A Dynamical Systems Approach

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A Dynamical Systems Approach to ModelingPlankton Food WebJulie M. [email protected] of MathematicsAlexei I. [email protected] of Electrical and Computer EngineeringGeorgia Institute of TechnologyMarch, 200511 AbstractThe main fo cus of this project is modeling phytoplankton predator-preysystems involving a resource, a prey (phytoplankton), and predators (zoo-plankton) in order to understand the complex interactions between these con-stituents. It is important to study these food chains, because phytoplanktoncontribute to numerous biogeochemical processes in nature. They controlwater quality, influence global climate by regulating carbon dioxide uptake,and form the basis for most aquatic food chains. Because phytoplankton aresimple and small organisms with short life spans, they are relatively easy tostudy. Lab and field experiments experiments can be readily combined withtheoretical analyses.In this paper, we first investigate a food web consisting of a chain ofa single zooplankton feeding on a phytoplankton which is dependent on aresouce. The model includes mathematically convenient approximation forthe fluctuation in resource availability and seasonal variations. We employ adynamical systems approach and supporting numerical simulations to studylong-term behavior. Our findings show that for long periods under forcing,distinct regimes of species coexistence are present and can be analyticallycomputed. As the total nutrient content is varied, the system undergo esseveral bifurcations, resulting in drastic, dynamical changes ranging fromcoexistence of both species to dominance of only zooplankton to extinctionof both. A smo oth seasonal transition, convenient for analytics, was alsointroduced allowing for the decline of resources not to be abrupt. A foodweb consisting of a resource, a phytoplankton, and two zooplankton speciescompeting for a single phytoplankton was also investigated. Most combina-2tions of parameters result in the dominance of one species of the zooplanktonwith total extinction of the other, which reduces to the first case examined.Using a type II functional response and considering only a narrow parametersubset, one is able to observe some interesting dynamics when switching ofzooplankton dominance occurs. We found that the length of the growing sea-son is directly involved in the system dynamics. The length of the growingseason can be too short for the phytoplankton or the fast growing zooplank-ton to emerge or it can b e long enough for the good competitor zooplanktonto dominate.2 IntroductionPhytoplankton contribute to numerous biogeochemical processes andconstitute the basis for most aquatic food chains. They play an essential rolein controlling water quality and exert a great influence on the global climateby regulating carbon dioxide (CO2) uptake.3Moreover, changes in phyto-plankton population can alert scientists to alterations in the environment.Phytoplankton can also be useful in determining where ocean currents pro-vide nutrients for plant growth and where pollutants poison the ocean andprevent plant growth.2Because phytoplankton are small, elementary crea-tures with short life spans, their dynamics are relatively easy to mo del. Inaddition, laboratory and field experiments are feasible.4, 5Phytoplankton require only a few things to survive: sunlight, water, andnutrients. Chlorophyll allows phytoplankton to utilize light energy, which isused to fix CO2to sugars and generate ATP.6Oxygen is then released as3a byproduct. Essential nutrients include CO2, nitrogen, sulfur, phosphoruscompounds, Si, Fe, and other trace metals. Some species of phytoplanktonalso require vitamins like thiamin or biotin to survive.4Predator-prey mo dels help forecast population trends and disease out-breaks. They can also aid in the understanding of biological communitystructure. The most basic kind of food chain involves a prey species feedingoff a nutrient. This can be generated to more complex models that includesp ecies interactions. The first predator-prey model was formulated by Lotkaand Volterra. The Lotka-Volterra system consists of one predator species andone prey species. Lotka-Volterra assumes the predator is completely depen-dent on the prey for its food supply and that the size of the prey populationis restricted only by predation. This model shows that predators and preycan coexist, in that both species follow stable cycle oscillations. Rosenzweigand MacArthur expanded this model further to include consumption ratesof predators.1Most predator-prey models do not include seasonal succession, or periodicresetting of system dynamics, and resource variations. Seasonal forcing isnecessary to better characterize phytoplankton development, because of thechanging physical environment. In the spring, light is a limiting resourcefor the phytoplankton in a lake. The water is warming and even mixingoccurs due to the lake’s wind and water currents. However, in the beginningstages of summer, when the phytoplankton start to grow, mixing occursmostly in two sections: the epilimnion, or warm layer of the lake, and thehypolimnion, or cold layer of the lake. There is also limited mixing betweeenlayers. Later in the summer, the nutrients become limited due to the now-4large biomass of phytoplankton and – as the epilimnion and the hypolimnionapproach the same temperature – there is greater mixing between layers.As the water temperature increases during the fall overturn, phytoplanktonget pulled to the b ottom when colder water sinks and nutrients rise to thesurface. Once the phytoplankton reach the bottom (cold) layer of the lake,it becomes almost impossible for the phytoplankton to resurface, and canobtain enough light and nutrients to survive. Thus, most phytoplankton dieoff in the fall overturn.The main question of this research effort is how does seasonal forcing or,more specifically, the length of the growing season affect system dynamics.The rest of this paper is organized as follows. First, we consider a system thatincludes a resource, one phytoplankton, and one zooplankton with seasonalvariation in resource availability. Subsequently, we examine a more complexmodel that includes a resource, one phytoplankton, and two zooplankton.3 Seasonal Forcing in a Phytoplankton FoodChain with One Zooplankton SpeciesPeriodic forcing can modeled mathematically by having one equation forthe growing season and a separate one for the dying


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