Study of Increasing Population and their Sensitivity Analysis with Second Order Differential Equations


  • Abha Singh


Single-species, Second-order differential equation model, stochastic stability, intraspecific interaction, age distribution, population model, living organism.


Models of populations are helpful tools for investigating populations in which intraspecific interactions account for the vast bulk of the variation in the population dynamics. These models are predicated on the assumption that we can learn about a species' dynamics simply by observing it. The growth dynamics of a homogeneous population is often described using first-order differential equations (both linear and nonlinear). Second, the paper attempts to develop a second-order differential equation model of single-species population growth that is consistent with Newtonian mechanics, and third, it attempts to investigate the stochastic stability of the population system to consider environmental fluctuation in the system. In this study, we provide a second-order differential equation model of a single species' population expansion that is consistent with Newtonian physics and evolutionary theory. Clark tested the validity of this hypothesis by collecting observational data from the bighorn sheep and mule deer populations in the American West and Southwest. As a result of the stochastic stability of the deterministic model, it is more probable that the model will be right even when the random environment changes somewhat.


Download data is not yet available.


Abbott, K. & Ives, A. (2012). Single-Species Population Models. In A. Hastings & L. Gross (Ed.), Encyclopedia of Theoretical Ecology (pp. 641-648). Berkeley: University of California Press.

Akçakaya, H. R., Ginzburg, L. R., Slice, D., & Slobodkin, L. B. (1988). The theory of population dynamics—II. Physiological delays. Bulletin of Mathematical Biology, 50(5), 503-515.

Anstead, K., Drew, K., Chagaris, D., Cieri, M., Schueller, A., McNamee, J., et al. (2020). The Past, Present, and Future of Forage Fish Management: a Case Study of Atlantic Menhaden. Front. Mar. Sci. Vol..

Arnold L. Stochastic differential equations. New York. 1974.

Banks RB. Growth and diffusion phenomena: Mathematical frameworks and applications. Springer Science & Business Media; 1993 Dec 22.

Bentley, J. W., Hines, D. E., Borrett, S. R., Serpetti, N., Hernandez-Milian, G., Fox, C., et al. (2019a). Combining scientific and fishers’ knowledge to cocreate indicators of food web structure and function. ICES J. Mar. Sci. 76, 2218–2234. doi: 10.1093/icesjms/fsz121.

Bentley, J. W., Serpetti, N., Fox, C. J., Heymans, J. J., and Reid, D. G. (2020). Retrospective analysis of the influence of environmental drivers on commercial stocks and fishing opportunities in the Irish Sea. Fisheries Oceanogr. 29, 415– 435. doi: 10.1111/fog.12486.

Bentley, J. W., Serpetti, N., Fox, C., Heymans, J. J., and Reid, D. G. (2019b). Fishers’ knowledge improves the accuracy of food web model predictions. ICES J. Mar. Sci. 76, 897–912. doi: 10.1093/icesjms/fsz003.

Clark, J.D. the second order derivative and population moddeling . Ecology, 52(1971), 606.

Drew, K., Cieri, M., Schueller, A., Buchheister, A., Chagaris, D., Nesslage, G., et al. (2020). Balancing Model Complexity, Data Requirements, and Management Objectives in Developing Ecological Reference Points for Atlantic Menhaden. Front. Mar. Sci. Vol

Ginzburg, L. R. (1987). The theory of population dynamics: back to first principles. In Mathematical Topics in Population Biology, Morphogenesis and Neurosciences (pp. 70-79). Springer, Berlin, Heidelberg.

Gardner, R. H., O'neill, R. V., Mankin, J. B., & Carney, J. H. (1981). A comparison of sensitivity analysis and error analysis based on a stream ecosystem model. Ecological Modelling, 12(3), 173-190.

Howell, D., Schueller, A. M., Bentley, J. W., Buchheister, A., Chagaris, D., Cieri, M., ... & Townsend, H. (2021). Combining ecosystem and single-species modeling to provide ecosystem-based fisheries management advice within current management systems. Frontiers in Marine Science, 7, 1163.

Innis, G. (1972). The second derivative and population modeling: another view. Ecology, 53(4), 720-723.

ICES (2020a). Baltic Fisheries Assessment Working Group (WGBFAS). ICES Sci. Rep. 2:643. doi: 10.17895/

ICES (2020b). Working Group on the Assessment of Demersal Stocks in the North Sea and Skagerrak (WGNSSK). ICES Sci. Rep. 2:1140. doi: 10.17895/ 6092

ICES (2020c). Workshop on an Ecosystem Based Approach to Fishery Management for the Irish Sea (WKIrish6; outputs from 2019 meeting). ICES Sci. Rep. 2:32. doi: 10.17895/

Mazo, R. M. (1981). On the Brownian motion of a frequencymodulated oscillator. Journal of Statistical Physics, 24(1), 39-44.

Mohler, L. L., Wampole, J. H., & Fichter, E. (1951). Mule deer in Nebraska National Forest. The Journal of Wildlife Management, 15(2), 129-157.

Rosen, R. J. (Ed.). (2013). Theoretical Biology and Complexity: three essays on the natural philosophy of complex systems. Academic Press.

Singh, A., Khan, R. A., Kushwaha, S., & Alshenqeeti, T. (2022). Roll of Newtonian and Non-Newtonian Motion in Analysis of Two-Phase Hepatic Blood Flow in Artery during Jaundice. International Journal of Mathematics and Mathematical Sciences, 2022.

Woodgerd, W. (1964). Population dynamics of bighorn sheep on Wildhorse Island. The Journal of Wildlife Management, 381-391.




How to Cite

Abha Singh. (2024). Study of Increasing Population and their Sensitivity Analysis with Second Order Differential Equations. International Journal of Intelligent Systems and Applications in Engineering, 12(22s), 49–53. Retrieved from



Research Article

Similar Articles

You may also start an advanced similarity search for this article.