By Kwadwo Antwi-Fordjour
Figure 1. Graph of the function \(g(u) = c u^p\), where \(c=1\) with varying \(p\). The green curve is for \(p=0.2\), the magenta curve is for \(p=0.5\), the blue curve is for \(p=0.8\), and the purple curve is for \(p=1\). Figure courtesy of Kwadwo Antwi-Fordjour.
Interdisciplinary research can provide solutions to challenging problems in multiple disciplines. Students therefore benefit from exposure to high-level undergraduate research in ecology that utilizes both mathematical and biological tools and concepts. Samford University in Birmingham, Ala., is a predominantly undergraduate institution that promotes undergraduate research in all programs. Two undergraduate students—Stephanie Westaway and Hannah Thompson—were heavily involved in my recent ecology research project, for which we also collaborated with Rana Parshad of Iowa State University. This work is currently under review in a peer-reviewed journal.
Previous studies in ecology and biology have extensively observed the indirect effects of predation that stem from fear of predators. Several experiments on vertebrates have confirmed that fear of predation leads to a decline in the prey species’ birth rate. The first mathematical model to describe the effect of fear in predator-prey dynamics, which emerged in 2016, multiplies the birth rate by a decreasing function of fear [4]. There is overwhelming scientific evidence that fear of predators can have both a stabilizing and destabilizing effect on populations within an ecosystem [2, 3].
Figure 2. One-parameter bifurcation plots that depict the changes in stability as the level of the fear parameter \((k)\) varies. 2a. There is a continuous decrease in both populations as the strength of fear parameter \((k)\) increases. We also observe a Hopf point \((H)\). The predator curve always lies above the prey curve as \(k\) changes from \(0\) to \(1\). 2b. We observed that as \(k\) increases from \(0\) to \(30\), the prey curve lies above the predator curve. We also observe the Hopf point \((H)\) here. Figure courtesy of [1].
Westaway and Thompson helped formulate a mathematical model that incorporates the combined effects of fear of predators, predator interference, and prey herd behavior. As the predator population increases, predator mutual interference hinders a predator’s searching efficiency. This intra-specific interaction among predators is a key factor for understanding the system’s dynamics. Herd behavior among prey species is a widely observed phenomenon in ecology. We modeled this phenomenon via the functional \(g(u) = c u^p\), where \(c\) is the rate of predation, \(u\) is the population density of the herd, and \(p\) is assumed to be between \(0\) and \(1\). This range for \(p\) means that the function is not smooth, which yields some rich and interesting dynamics in the model — particularly, the possibility of finite-time extinction of the species. Figure 1 illustrates the shape of the functions that describe the herd behavior of the population. The most common function to depict herd behavior is the square root function—i.e., \(p=0.5\)—which Georgy F. Gause proposed in 1934.
Figure 3. One-parameter bifurcation plot that depicts saddle-node bifurcation. Figure courtesy of [1].
The students numerically computed the equilibrium points with the help of
Mathematica, a mathematical software. They demonstrated the existence of the trivial or extinction equilibrium point, the axial or boundary equilibrium point, and the coexistence equilibria. Using concepts from linear algebra and differential equations, Westaway and Thompson observed that a strong strength of fear can stabilize an unstable coexistence equilibrium point. In contrast, a weak strength of fear may destabilize a stable coexistence equilibrium point under certain biologically feasible parameter restrictions. The local changes in stability that transpire as the level of fear fluctuates are described by Hopf bifurcation, which we corroborated via MATLAB’s
MatCont package (see Figure 2). While experimenting with a hypothetical parameter set, Westaway and Thompson made another important observation: the possibility of a saddle-node bifurcation. This type of bifurcation occurs when a change in a parameter value leads to the disappearance of two contrasting equilibrium points after they collide. The point of collision between these two points is called the limit point or saddle-node point. Figure 3 clearly illustrates this phenomenon with the aid of the MatCont package.
In conclusion, the interplay between the effects of fear of predators, mutual interference, and prey herd behavior provides several rich dynamical outcomes [1]. One interesting application of fear occurs in epidemiology, and there are several ongoing investigations in this field. Westaway and Thompson had a great experience with this research; they presented their findings at seminars and local conferences and now feel motivated to pursue graduate studies in this area. I am currently working with Zachary Overton, another undergraduate student at Samford, to extend this research to include the Allee effect in the prey species. Through this kind of interdisciplinary research, students develop strong analytical and computational skills in applied mathematics and related areas.
Kwadwo Antwi-Fordjour presented this research during a minisymposium presentation at the 2022 SIAM Conference on Applied Mathematics Education (ED22), which took place concurrently with the 2022 SIAM Annual Meeting in Pittsburgh, Pa., this July. He received funding to attend ED22 through a SIAM Early Career Travel Award. To learn more about Early Career Travel Awards and submit an application, visit the online page.
Acknowledgments: Registration and travel support for Antwi-Fordjour was provided by the National Science Foundation (NSF grant DMS – 1757085). This research is supported by a Samford Faculty Development Grant (FDG084).
References
[1] Antwi-Fordjour, K., Parshad, R.D., Thompson, H.E., & Westaway, S.B. (2021). Fear-driven extinction and (de)stabilization in a predator-prey model incorporating prey herd behavior and mutual interference. Preprint, arXiv:2108.00546v1.
[2] Kim, S., & Antwi-Fordjour, K. (2022). Prey group defense to predator aggregate induced fear. Eur. Phys. J. Plus, 137(704), 1-17.
[3] Verma, H., Antwi-Fordjour, K., Hossain, M., Pal, N., Parshad, R.D., & Mathur, P. (2021). A “double” fear effect in a tri-trophic food chain model. Eur. Phys. J. Plus, 136(905), 1-17.
[4] Wang, X., Zanette, L., & Zou, X. (2016). Modelling the fear effect in predator-prey dynamics. J. Math. Biol., 73, 1179-1204.
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Kwadwo Antwi-Fordjour is an applied mathematician. He is originally from Ghana and is currently an assistant professor of mathematics at Samford University. |