Modeling approaches to understand plant-pollinator-herbivore interactions

Plant-pollinator interactions play an important role in the maintenance of the balance of nature. All organisms living in the environment are composed of different ratios of chemical elements. By considering the balance of essential chemical elements in nature, we can formulate mathematical models to study their role in the dynamics of the system as well as nature. We formulate and analyze stoichiometric plant-pollinator and stoichiometric herbivore-plant-pollinator models. Our models include three dimensional and four-dimensional systems of ordinary differential equations to represent the plant, pollinator, herbivore populations, as well as the varying nutrient levels of the plant. We analyze the dynamics of the systems such as non-negativeness and boundedness of solutions, as well as the existence and stability of boundary equilbria. We use numerical simulations, parameter sensitivity analyses (Latin hypercube sampling (LHS) with the partial rank correlation coefficient technique), and bifurcation analyses to explore model dynamics. LHS parameter sensitivity analyses show that the search rate and the carrying capacity of pollinators are the most important parameters to the stoichiometric plant-pollinator model. Numerical simulations and bifurcation analysis show the existence of critical thresholds of the number of pollinators for the persistence of plants and herbivores.

Environmental changes are one of the most evident features which have important impacts on populations living in nature. We incorporate environmental seasonal into our plant-pollinator and plant-pollinator-herbivore models using periodic functions. Here we add a seasonal variation term to pollinator carrying capacity and investigate the new seasonal system. In addition to the studies of non-negativeness and boundedness of the models solutions, numerical and bifurcation analyses are performed. These analyses show that environmental seasonality can play a role in shifting the threshold of pollinators needed for plant and herbivore populations to survive.

Different species behaviours in ecological interactions have been studied recently using several approaches. Among those approaches, many researchers have studied Holling type functional responses. We study our plant-pollinator-herbivore model by replacing the previously used Holling type II functional response with a Holling type IV functional response term to describe the interaction between plant and herbivore. Non-negativeness and boundedness of the solutions are proved. Numerical and bifurcation analyses show that the threshold of pollinators needed for plants to survive remains almost the same as the original plant-pollinator-herbivore model.