Modeling the Effects of Fruit Harvesting with an Application to Pentadesma Trees

By Benito Chen-Charpentier

Plants are essential to life on Earth. They provide food, shelter, and oxygen to both humans and animals; help prevent soil erosion; support more stable weather patterns; and provide countless substances for medicine, industry, and everyday life. But just like all living organisms, plants are subject to multiple factors that affect their sustainability. One major threat to plant survival is disease, which may be caused by pathogens like viruses, fungi, and bacteria; environmental conditions; and even parasitic plants. Yet plants also face many other hazards, such as insect plagues, overharvesting, climate change, and especially habitat reduction. Road construction, forest fires, overharvesting, and agricultural and urban growth all contribute to habitat reduction and fragmentation. Species in reduced habitats are more susceptible to competition from invaders, decreased levels of diversity, diseases and plagues, reduced growth rates, and increased death rates. Of course, the harvesting of entire plants—including trees for wood and paper as well as shrubs, grasses, and other vegetation for clothing, shelter, and food—is an enormous threat to species sustainability and can lead to species extinction.

Understanding the effects of habitat reduction on plant species is quite important, but actually performing experiments can pose numerous difficulties. We therefore developed a mathematical model to study and simulate the dynamics of plants in different-sized habitats. Our model is based on ordinary differential equations (ODEs) for plant biomass and assumes that the net growth rate is dependent on the area of the habitat fragment in question. It accounts for intraspecies competition, which yields a logistic equation when the growth rate is positive. While limited experimental data exists on the way in which habitat size affects the population, data from an experiment with Heliconia acuminata in Brazil—which isolated plants in habitats of different areas—suggested the presence of a critical minimum area for species survival [1]. Accordingly, we fit the data with a simple rational function for the growth rate as a function of area and then determined the critical area.

However, plants that undergo habitat reduction also face other stressors. For instance, the harvesting of non-timber forest products (NTFPs) such as fruits, nuts, and leaves may have smaller effects than the harvesting of whole plants, but the consequences can still sometimes be drastic. One example is the harvesting of fruit from Pentadesma butyracea trees in Benin. Fruit harvesting decreases the number of seeds that are available for growth, leading to increases in clonal reproduction that consequently limit the species’ genetic diversity. Pentadesma trees are also subject to habitat reduction. In addition, the seeds of their fruit produce a butter that is used as a cooking oil and has applications in cosmetics and pharmaceuticals — which has increased its demand. Given the reduction of its habitat and the cutting of trees for timber, Pentadesma is becoming an endangered species in Benin as well as its neighboring countries.

Using existing data for the number of Pentadesma trees in differently sized plots of two climatic regions in Benin over several years [2], we generated a new ODE-based model to study the combined effects of habitat reduction and nonlethal harvesting of NTFPs. We sought to identify the nonlethal harvest rate and area that will ensure Pentadesma sustainability; we also aimed to examine the effect of climate variations—specifically humidity and rainfall—on these results. We utilized the limited available data to fit the model parameters and performed simulations for different harvesting rates, assuming that the harvesting rates were independent of time. The model includes the population density growth rate’s dependence on both habitat size and nonlethal harvesting rate of the fruit, thus enabling the analysis and simulation of plant population dynamics under two stresses. We applied our model to numerous areas within two different climatic regions. Since real-world experiments are not feasible, the model is particularly useful because it can help establish policies that will ultimately sustain the Pentadesma tree and encourage the rational harvesting of its fruit. 

Because the Pentadesma fruit is a valuable commodity, we must develop harvesting rates that will maximize the economic benefit for local harvesting communities while also maintaining tree viability. We hence used optimal control methods to develop nonlethal fruit harvesting policies in order to identify the strategies that maximize economic benefit (net profit). Pentadesma fruit is collected from the ground, and we consider a single harvest season that typically lasts about three months. The rate of collection is a function of time, since the number of fruits on the ground is time dependent. The objective function to be maximized is the total value of the collected fruits minus the harvesting cost, which is proportional to the square of the harvesting rate divided by the total number of fruits on the ground. Here, we assume that the cost of fruit collection depends on the square of the collection rate and gets increasingly more expensive as the number of fruits decreases. This setup has two constraints: the ODE model’s dependence on the area and harvesting rate, and the fact that the amount of harvested fruit must be positive. It also accounts for the inevitable reality that some fruit will be stolen. 

We performed the optimization for different scenarios and changed the cost of fruit collection, the number of fruits on the ground, and the percentage of fruit stolen. Our model predicts the optimal harvesting rate, and the simulations can guide the harvesters. Note that only the cost of collecting the fruit depends on time, whereas the rest of the processing tasks (drying the fruit, extracting the seeds, and transporting them) is almost time independent.

Importantly, we found that it is possible to enact harvesting policies that maximize profit while maintaining a viable plant population. Researchers could easily modify our procedure for Pentadesma to create similar policies for other plants. 

Acknowledgments: This work was performed jointly with Folashade Agusto, Orou Gaoue, Natali Hritonenko, and Maria Leite.

References
[1] Bruna, E.M. (2003). Are plant populations in fragmented habitats recruitment limited? Tests with an Amazonian herb. Ecology, 84(4), 932-947.
[2] Gaoue, O.G., Gado, C., Natta, A.K., Kouagou, M. (2018). Recurrent fruit harvesting reduces seedling density but increases the frequency of clonal reproduction in a tropical tree. BioTrop., 50(1), 69-73.

Benito Chen-Charpentier holds a B.S. in physics from the National Autonomous University of Mexico and a Ph.D. in applied mathematics from the California Institute of Technology. He is currently a professor of mathematics at the University of Texas at Arlington. His areas of research include mathematical biology, mathematical modeling, and numerical analysis.