# American Institute of Mathematical Sciences

2018, 23(2): 493-508. doi: 10.3934/dcdsb.2017194

## A unifying approach to discrete single-species populations models

 Department of Mathematics and Statistics, Georgetown University, Washington, DC 20057-1233, USA

* Corresponding author

Received  February 2017 Revised  April 2017 Published  December 2017

Fund Project: I wish to thank the reviewers for their helpful suggestions

 $f$
for the discrete-time density-dependent population model
 $p_{n+1} =f(p_n)$
as
 $f(p) =p+r(p)p$
where
 $r$
is the per capita growth rate. Making reasonable assumptions about the intraspecies relationships for the population, we develop four conditions that the function
 $r$
should satisfy. We then analyze the implications of these conditions for the recruitment function
 $f$
. In particular, we compare our conditions to those of Cull [2007], finding that the Cull model, with two additional conditions, is equivalent to our model.
Studying the per capita growth rate when satisfying our four conditions gives insight into contest and scramble competition. In particular, depending on the properties of
 $r$
and
 $f$
, we have two different types of contest and scramble competitions, depending on the size of the population. We finally extend our approach to develop new models for discontinuous recruitment functions and for populations exhibiting Allee effects.
Citation: James Sandefur. A unifying approach to discrete single-species populations models. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 493-508. doi: 10.3934/dcdsb.2017194
##### References:
 [1] T. S. Bellows, The descriptive properties of some models for density dependence, Journal of Animal Ecology, 50 (1981), 139-156. doi: 10.2307/4037. [2] R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Fish & Fisheries Series, 1993. doi: 10.1007/978-94-011-2106-4. [3] P. Cull, Population models: Stability in one dimension, Bulletin Mathematical Biology, 69 (2007), 989-1017. doi: 10.1007/s11538-006-9129-1. [4] M. P. Hassell, Density dependence in single-species populations, Journal of Animal Ecology, 44 (1975), 283-295. doi: 10.2307/3863. [5] R. J. Higgins, C. M. Kent, V. L. Kocic and Y. Kostrov, Dynamics of a nonlinear discrete population model with jumps, Applicable Analysis and Discrete Mathematics, 9 (2015), 245-270. doi: 10.2298/AADM150930019H. [6] V. L. Kocic and Y. Kostrov, Dynamics of a discontinuous discrete Beverton-Holt model, Journal of Difference Equations and Applications, 20 (2014), 859-874. doi: 10.1080/10236198.2013.824968. [7] M. Liermann and R. Hilborn, Depensation: Evidence, models and implications, Fish and Fisheries, 2 (2001), 33-58. [8] J. Maynard-smith and M. Slatkin, The stability of predator-prey systems, Ecology, 54 (1973), 384-391. [9] R. M. May, Simple mathematical models with very complicated dynamics, The Theory of Chaotic Attractors, (2004), 85-93. doi: 10.1007/978-0-387-21830-4_7. [10] W. E. Ricker, Stock and recruitment, Journal of the Fisheries Research Board of Canada, 11 (1954), 559-623. doi: 10.1139/f54-039. [11] J. T. Sandefur, Discrete Dynamical Systems: Theory and Applications, Oxford University Press, Oxford, 1990. [12] M. Williamson, The analysis of discrete time cycles, Ecological stability (eds. M. B. Usher and Author), Chapman and Hall, (1974), 17–33. doi: 10.1007/978-1-4899-6938-5_2.

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##### References:
 [1] T. S. Bellows, The descriptive properties of some models for density dependence, Journal of Animal Ecology, 50 (1981), 139-156. doi: 10.2307/4037. [2] R. J. H. Beverton and S. J. Holt, On the Dynamics of Exploited Fish Populations, Fish & Fisheries Series, 1993. doi: 10.1007/978-94-011-2106-4. [3] P. Cull, Population models: Stability in one dimension, Bulletin Mathematical Biology, 69 (2007), 989-1017. doi: 10.1007/s11538-006-9129-1. [4] M. P. Hassell, Density dependence in single-species populations, Journal of Animal Ecology, 44 (1975), 283-295. doi: 10.2307/3863. [5] R. J. Higgins, C. M. Kent, V. L. Kocic and Y. Kostrov, Dynamics of a nonlinear discrete population model with jumps, Applicable Analysis and Discrete Mathematics, 9 (2015), 245-270. doi: 10.2298/AADM150930019H. [6] V. L. Kocic and Y. Kostrov, Dynamics of a discontinuous discrete Beverton-Holt model, Journal of Difference Equations and Applications, 20 (2014), 859-874. doi: 10.1080/10236198.2013.824968. [7] M. Liermann and R. Hilborn, Depensation: Evidence, models and implications, Fish and Fisheries, 2 (2001), 33-58. [8] J. Maynard-smith and M. Slatkin, The stability of predator-prey systems, Ecology, 54 (1973), 384-391. [9] R. M. May, Simple mathematical models with very complicated dynamics, The Theory of Chaotic Attractors, (2004), 85-93. doi: 10.1007/978-0-387-21830-4_7. [10] W. E. Ricker, Stock and recruitment, Journal of the Fisheries Research Board of Canada, 11 (1954), 559-623. doi: 10.1139/f54-039. [11] J. T. Sandefur, Discrete Dynamical Systems: Theory and Applications, Oxford University Press, Oxford, 1990. [12] M. Williamson, The analysis of discrete time cycles, Ecological stability (eds. M. B. Usher and Author), Chapman and Hall, (1974), 17–33. doi: 10.1007/978-1-4899-6938-5_2.
Maynard-Smith/Slatkin model with $j =0.5$ and $b =2$. Recruitment function is in graph a), per capita recruitment in graph b), and per capita growth rate in graph c).
Maynard-Smith/Slatkin model with j =2 and b =2. Recruitment function is in graph a), per capita recruitment in graph b), and per capita growth rate in graph c).
Maynard-Smith/Slatkin recruitment function with j =2 and b =0.5.
Maynard-Smith/Slatkin model with $b =1.2$, and with $j =1.5$ for $p\leq 1$ and $j =0.5$ for $p>1$. Recruitment function in figure a) is increasing and per capita growth rate function in b) switches concavity.
Recruitment function (on left) and per capita growth rate function (on right) for model 15
Recruitment and per capita growth rate functions for Hassell model 14 with $b =2$ and $j =3$. Satisfies $C_1$ and $S_2$.
Discontinuous Beverton Holt model with $a_1 =1$, $a_2 =0.5$, $c_1 =0.5$, $c_2 =0.3$, $M_1 =1.5$, $M_2 =0.5$.
The Maynard-Smith/Slatkin discontinuous model 25.
The recruitment function (graph a)) for the per capita growth rate function 27 (graph b)) with $j =2$, $b =0.8$, and $m =0.6$. Minimum viable population size is $0.2$.
The recruitment function (graph a)) for the per capita growth rate function 28 (graph b)) with $j =0.7$, $b =0.8$, and $m =0.6$. Minimum viable population size is $0.2$.
graph a) is the recruitment function $F$ for 31. graph b) is the corresponding per capita growth rate function $r$.
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