Graduate Research

My primary research interests lie at the intersection of partial differential equations and mathematical biology. I specialize in investigating various qualitative properties of population dynamics models in ecology. Specifically, I focus on exploring the impact of fear on Lotka-Volterra-type competition models. Fear is a crucial factor in the behavior of many species, and it can have a significant effect on their competitive interactions. Through studying the role of fear in these models, I aim to gain insight into the complex dynamics of biological systems.

In addition to my work on competition models, I also explore the Chemotactic Diffusion System, a mathematical model that describes the spatiotemporal evolution of two biological species in a competitive scenario. This system is especially relevant to understanding the competition for limited resources between bacterial populations. By analyzing the dynamics of this system, I hope to gain further understanding of how different species interact and compete in real-world ecological settings.

Overall, my research contributes to advancing our understanding of the dynamics of biological populations and their responses to environmental factors. By developing mathematical models and exploring their properties, I aim to identify new mechanisms underlying ecological systems and predict their responses to perturbations.

☞ The effect of fear on two species competition

Non-consumptive effects such as fear of depredation, can strongly influence predator-prey dynamics. There are several ecological and social motivations for these effects in competitive systems as well. In this work we consider the classic two species ODE and PDE Lotka-Volterra competition models, where one of the competitors is fearful of the other. We find that the presence of fear can have several interesting dynamical effects on the classical competitive scenarios. Notably, for fear levels in certain regimes, we show novel bi-stability dynamics. Furthermore, in the spatially explicit setting, the effects of several spatially heterogeneous fear functions are investigated. In particular, we show that under certain integral restrictions on the fear function, a weak competition type situation can change to competitive exclusion. Applications of these results to ecological as well as sociopolitical settings are discussed, that connect to the landscape of fear (LOF) concept in ecology.

Relevant publications

[1] Srivastava, V., Takyi, E. M., & Parshad, R. D. The effect of "fear" on two species competition. Mathematical Biosciences and Engineering, 2023, 20(5): 8814-8855.
[PDF].

[2] Srivastava, V., Takyi, E. M., & Parshad, R. D. (2022). The effect of "fear" on two species competition. ArXiv preprint arXiv:2210.10280.
[PDF].

☞ Dynamical Analysis of a Lotka-Volterra Competition Model with both Allee and Fear Effect

Population ecology theory is replete with density dependent processes. However trait-mediated or behavioral indirect interactions can both reinforce or oppose density-dependent effects. This paper presents the first two species competitive ODE and PDE systems where an Allee effect, which is a density dependent process and the fear effect, which is non-consumptive and behavioral are both present. The stability of the equilibria is discussed analytically using the qualitative theory of ordinary differential equations. It is found that the Allee effect and the fear effect change the extinction dynamics of the system and the number of positive equilibrium points, but they do not affect the stability of the positive equilibria. We also observe some special dynamics that induce bifurcations in the system by varying the Allee or fear parameter. Interestingly we find that the Allee effect working in conjunction with the fear effect, can bring about several qualitative changes to the dynamical behavior of the system with only the fear effect in place, in regimes of small fear. That is, for small amounts of the fear parameter, it can change a competitive exclusion type situation to a strong competition type situation. It can also change a weak competition type situation to a bi-stability type situation. However for large fear regimes the Allee effect reinforces the dynamics driven by the fear effect. The analysis of the corresponding spatially explicit model is also presented. To this end the comparison principle for parabolic PDE is used. The conclusions of this paper have strong implications for conservation biology, biological control as well as the preservation of biodiversity.

Relevant publications

[1] Chen, S., Chen, F., Srivastava, V., & Parshad, R. D. (2023). Dynamical Analysis of a Lotka-Volterra Competition Model with both Allee and Fear Effect. ArXiv preprint arXiv:2303.04919.
[PDF].

☞ Brownian dynamics simulations for the narrow escape problem in the unit sphere

This work focuses on the narrow escape problem of calculating the time needed for a Brownian particle to leave a domain with localized absorbing boundary traps. The mean first-passage time (MFPT) is modeled by a Poisson partial differential equation and the objective is to perform direct numerical simulations of multiple particles and compare the results with the Poisson equation-based continuum model. The simulations validate the model and also provide additional insights into particle dynamics that cannot be captured in a continuum approach. The Poisson equation-based continuum model's validation is achieved through 10⁴ particle runs. Additional insights into particle dynamics, such as isotropic vs. anisotropic diffusion effects, are gained through direct simulations.

Relevant publications

[1] Srivastava, V., & Cheviakov, A. (2021). Brownian dynamics simulations for the narrow escape problem in the unit sphere. Physical Review E, 104(6), 064113.
[Link]

[2] Srivastava, V., & Cheviakov, A. (2021). Narrow Escape Brownian Dynamics Modeling in the Three-Dimensional Unit Sphere. ArXiv preprint arXiv:2107.01233.
[PDF].

[3]Poster

☞ Master's Thesis

For my thesis, I investigated how strongly localized perturbations affect the solution of the eigenvalue problem, which has applications in understanding physical and biological phenomena such as the Narrow Escape problem. Specifically, I focused on analyzing the eigenvalues and eigenfunctions of this problem, with a particular interest in properties such as isolatedness, regularity, and simplicity that are associated with these eigenpairs.

Relevant publications

[1] Srivastava, V. The Qualitative Study of the Eigenvalue and Eigenfunctions of the Strong Localized Perturbed Eigenvalue Problem (A dissertation submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics), (2020).