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# A unifying approach to discrete single-species populations models

• * Corresponding author

I wish to thank the reviewers for their helpful suggestions

• In this article, we write the recruitment function $f$ 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 , 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.

Mathematics Subject Classification: Primary: 92D25; Secondary: 39A60.

 Citation: • • Figure 1.  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).

Figure 2.  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).

Figure 3.  Maynard-Smith/Slatkin recruitment function with j =2 and b =0.5.

Figure 4.  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.

Figure 5.  Recruitment function (on left) and per capita growth rate function (on right) for model 15

Figure 6.  Recruitment and per capita growth rate functions for Hassell model 14 with $b =2$ and $j =3$. Satisfies $C_1$ and $S_2$.

Figure 7.  Discontinuous Beverton Holt model with $a_1 =1$, $a_2 =0.5$, $c_1 =0.5$, $c_2 =0.3$, $M_1 =1.5$, $M_2 =0.5$.

Figure 8.  The Maynard-Smith/Slatkin discontinuous model 25.

Figure 9.  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 $0.2$.

Figure 10.  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 $0.2$.

Figure 11.  graph a) is the recruitment function $F$ for 31. graph b) is the corresponding per capita growth rate function $r$.

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