From nest site lottery to host lottery: continuous model of growth suppression driven by the availability of nest sites for newborns or hosts for parasites and its impact on the selection of life history strategies

Abstract

The idea that selection works in different ways during free population growth and at the equilibrium population size has been present in theoretical biology for a long time. It was first expressed as an r and K selection concept and later clarified in the debate on fitness measures in life history theory. The latest discussion related to this topic is focused on the nest site lottery mechanism and the resulting new population growth model. In this mechanistic biphasic model, the suppression of growth is induced by a shortage of free nest sites for newborns. Before it occurs, the population can grow exponentially. In this paper, the continuous version of the model and its selective properties are analysed. We show a continuous smooth transition between different fitness measures operating during the exponential growth and suppressed growth phase and at the equilibrium population size. Then, the model is extended to the case of a population of parasites, where a constant number of nest sites is replaced by the dynamics of a population of their hosts, in the role of the limiting supply. Parasite strategies are selected under exponential and suppressed growth phases of the population of hosts. Transitions between different fitness measures and conditions for extinction of hosts by parasites are analysed. An interesting result is the possibility of a continuum of fitness measures of parasites for the unsuppressed exponential growth of the host population.

Appendices

Appendix 1: Proof of Theorem 1

Let us start from subsystem (29)

$$\begin{aligned} {\dot{q}}_{i}(t)=q_{i}(t)\left[ \left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right) \left( \dfrac{K}{n(t)}-1\right) /\tau -\left( d_{i}-{\bar{d}}(t)\right) \right] . \end{aligned}$$

Let us substitute $\left( \dfrac{K}{n(t)}-1\right) /\tau ={\bar{d}}(t)+\alpha$ , which implies $\alpha =\dfrac{K-n(t)}{n(t)\tau }-{\bar{d}}(t)$. This leads to

$$\begin{aligned} {\dot{q}}_{i}(t)&=q_{i}(t)\left[ \left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right) \left( \dfrac{K}{n(t)}-1\right) /\tau -\left( d_{i}-{\bar{d}}(t)\right) \right] = \nonumber \\&=q_{i}(t)\left[ \left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right) {\bar{d}} (t)-\left( d_{i}-{\bar{d}}(t)\right) +\left( \dfrac{b_{i}}{{\bar{b}}(t)} -1\right) \alpha \right] = \nonumber \\&=q_{i}(t)\left[ \left( b_{i}\dfrac{{\bar{d}}(t)}{{\bar{b}}(t)}-{\bar{d}} (t)\right) -\left( d_{i}-{\bar{d}}(t)\right) \right. \nonumber \\&\left. \quad +\left( \dfrac{b_{i}}{{\bar{b}}(t)} -1\right) \left( \dfrac{K-n(t)}{n(t)\tau }-{\bar{d}}(t)\right) \right] \nonumber \\&=q_{i}(t)\left[ \left( b_{i}\dfrac{{\bar{d}}(t)}{{\bar{b}}(t)}-d_{i}\right) +\left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right) \left( \dfrac{K-n(t)}{n(t)\tau }- {\bar{d}}(t)\right) \right] . \end{aligned}$$

(58)

The last bracketed term can be presented in the following form:

$$\begin{aligned} \left( \dfrac{K-n(t)}{n(t)\tau }-{\bar{d}}(t)\right)&=\dfrac{K-n(t)-{\bar{d}} (t)n(t)\tau }{n(t)\tau }=\dfrac{K-n(t)\left( 1+{\bar{d}}(t)\tau \right) }{ n(t)\tau } \\&=\dfrac{1+{\bar{d}}(t)\tau }{n(t)\tau }\left( \dfrac{K}{1+{\bar{d}}(t)\tau } -n(t)\right) . \end{aligned}$$

Let us substitute the density attractor ${\tilde{n}}(t)=\dfrac{K}{\tau{\bar{d}} (t)+1 }$ in the above formula. This will lead to the form being a function of the distance from the stable density manifold measured by the bracketed term:

$$\begin{aligned} \dfrac{1+{\bar{d}}(t)\tau }{n(t)\tau }\left( {\tilde{n}}(t)-n(t)\right) =\left( \dfrac{{\tilde{n}}(t)}{n(t)}-1\right) \left( 1/\tau +{\bar{d}}(t)\right) . \end{aligned}$$

In effect, we obtain the following system:

$$\begin{aligned} {\dot{q}}_{i}(t)&=q_{i}(t)\left[ \left( b_{i}\dfrac{{\bar{d}}(t)}{{\bar{b}}(t)} -d_{i}\right) \right. \\&\left. \quad +\left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right) \left( \dfrac{ {\tilde{n}}(t)}{n(t)}-1\right) \left( 1/\tau +{\bar{d}}(t)\right) \right] , \\ {\dot{n}}(t)&=n(t)\left( \dfrac{K-n(t)}{n(t)\tau }-{\bar{d}}(t)\right) , \end{aligned}$$

where the factor

$$\begin{aligned} \left( b_{i}\dfrac{{\bar{d}}(t)}{{\bar{b}}(t)}-d_{i}\right) =d_{i}\left( \dfrac{ b_{i}}{d_{i}}\dfrac{{\bar{d}}(t)}{{\bar{b}}(t)}-1\right) =d_{i}\left( \dfrac{ L_{i}}{{\bar{L}}(t)}-1\right) \end{aligned}$$

is compatible with the nest site lottery mechanism (which implies maximization of $L_{i}=b_{i}/d_{i}$ and $d_{i}$ among strategies with maximal $L_{i}$). Let us consider the second factor

$$\begin{aligned} \left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right) \left( \dfrac{{\tilde{n}}(t)}{n(t)} -1\right) \left( 1/\tau +{\bar{d}}(t)\right) , \end{aligned}$$

where the term $\left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right)$, responsible for the maximization of $b_{i}$, and the term $\left( \dfrac{{\tilde{n}}(t)}{n(t)} -1\right)$ vanish with convergence to the nullcline ${\tilde{n}}(t)$. After the substitution of the switching point ${\hat{n}}=\dfrac{K}{\tau {\bar{b}}+1}$ as the population size, the replicator dynamics (29) reduce to the unsuppressed case ${\dot{q}}_{i}(t)=(b_{i}-{\bar{b}}(t))-\left( d_{i}-{\bar{d}} (t)\right)$. This means that the decrease in the coefficient

$$\begin{aligned} \left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right) \left( \dfrac{{\tilde{n}}(t)}{n(t)} -1\right) \left( 1/\tau +{\bar{d}}(t)\right) \end{aligned}$$

is responsible for the continuous transition from unsuppressed exponential growth to the nest site lottery mechanism.

Appendix 2: Proof of Theorem 2

From (39)–(40), we have

$$\begin{aligned} {\dot{x}}(t)&=\frac{{\dot{n}}(t)K(t)-n(t){\dot{K}}(t)}{K^{2}(t)} \\&=\frac{1}{K^{2}(t)}\left[ \left( \frac{K(t)}{\tau }-\left[ \frac{1}{\tau } +d+D+D_{p}\right] n(t)\right) K(t) \right. \\&\quad \left. -n(t)\left( \left[ B-D\right] K(t)-D_{p}n(t)\right) \right] \\&=\frac{1}{K^{2}(t)}\left[ \frac{K^{2}(t)}{\tau }-\left[ \frac{1}{\tau } +d+D_{p}+B\right] n(t)K(t)+D_{p}n^{2}(t)\right] \\&=\frac{1}{\tau }-\left[ \frac{1}{\tau }+d+D_{p}+B\right] x(t)+D_{p}x^{2}(t). \end{aligned}$$

Thus, the function x(t) is a solution of the differential equation ${\dot{x}}(t)=P(x(t))$, where $P(x)=\tau ^{-1}-ax+D_{p}x^{2}$ and $a=\tau ^{-1}+d+D_{p}+B$. Note that in the above derivation, D cancels out. Therefore, every other constant or function added to D will also vanish. This means that x is independent of all host mortality factors. Since

$$\begin{aligned} \Delta =a^{2}-4D_{p}\tau ^{-1}>(\tau ^{-1}+D_{p})^{2}-4D_{p}\tau ^{-1}=(\tau ^{-1}-D_{p})^{2}\ge 0 \end{aligned}$$

we find that $\Delta >0$, and the quadratic polynomial P(x) has two positive roots:

$$\begin{aligned} x_{*} & = \frac{a-\sqrt{a^{2}-4D_{p}\tau ^{-1}}}{2D_{p}}, \\ x^{*} & = \frac{a+\sqrt{a^{2}-4D_{p}\tau ^{-1}}}{2D_{p}}. \end{aligned}$$

The points $x_{*}$ and $x^{*}$ are, respectively, stable and unstable stationary solutions of the equation ${\dot{x}}(t)=P(x(t))$. We have $x_{*}<1$. Indeed, $x_{*}<1$ if $a-2D_{p}<\sqrt{ a^{2}-4D_{p}\tau ^{-1}}$, leading to $a>D_{p}+\tau ^{-1}$, which is obviously always satisfied. Furthermore, from $a>D_{p}+\tau ^{-1}$, we find that $2D_{p}-a<\sqrt{a^{2}-4D_{p}\tau ^{-1}}$, which yields $x^{*}>1$. Recall that by definition, $n\le K$; thus, we have that $x\le 1$. Therefore, $x^{*}$ can be rejected as nonbiological and x(t) converges to $x_{*}$ as $t\rightarrow \infty$.

Appendix 3: Proof of Theorem 3

We know that trajectories can converge to (0, 0) or escape to infinity. System (38)–(40) can be presented in the form

$$\begin{aligned} {\dot{n}}(t)&=n(t)(b-d-D-D_{p}), \nonumber \\&\quad {\hbox {when}}\,\, n(t)\tau b \le K(t)-n(t), \end{aligned}$$

(59)

$$\begin{aligned} {\dot{n}}(t)&=\dfrac{K(t)}{\tau }-n(t)\left( \dfrac{1}{\tau } +d+D+D_{p}\right) , \nonumber \\&\quad {\hbox {when}}\,\, n(t)\tau b >K(t)-n(t). \end{aligned}$$

(60)

$$\begin{aligned} {\dot{K}}(t)&=K(t)(B-D)-D_{p}n(t). \end{aligned}$$

(61)

If condition (42) is satisfied, then we enter the suppression phase when $n(t)\tau b>K(t)-n(t)$ and remain there. This means that for a sufficiently large time, we have the system

$$\begin{aligned} {\dot{n}}(t)&=\dfrac{K(t)}{\tau }-n(t)\left( \dfrac{1}{\tau } +d+D+D_{p}\right) , \end{aligned}$$

(62)

$$\begin{aligned} {\dot{K}}(t)&=K(t)(B-D)-D_{p}n(t). \end{aligned}$$

(63)

The last system is linear, and the behaviour of its solutions depends on the eigenvalues of the matrix

$$\begin{aligned} A= \begin{pmatrix} -\alpha & \tau ^{-1} \\ -D_{p} & B-D \end{pmatrix} ,\quad \alpha =\tau ^{-1}+d+D+D_{p}. \end{aligned}$$

The eigenvalues of A satisfy the following equations:

$$\begin{aligned} (\lambda +\alpha )(\lambda +D-B)+D_{p}\tau ^{-1}=0, \end{aligned}$$

equivalent to

$$\begin{aligned} \lambda ^{2}+(\alpha +D-B)\lambda +\alpha (D-B)+D_{p}\tau ^{-1}=0. \end{aligned}$$

We have

$$\begin{aligned} \Delta&=(\alpha +D-B)^{2}-4\alpha (D-B)-4D_{p}\tau ^{-1} \\&=(\alpha +B-D)^{2}-4D_{p}\tau ^{-1} \\&>(\tau ^{-1}+D_{p})^{2}-4D_{p}\tau ^{-1}=(\tau ^{-1}-D_{p})^{2}\ge 0. \end{aligned}$$

This means that the eigenvalues of A are real numbers and the solutions of (62)–(63) are bounded if and only if $\lambda _{1}\le 0$ and $\lambda _{2}\le 0$, which holds if

$$\begin{aligned} \alpha +D-B\ge 0 \text { and }\alpha (D-B)+D_{p}\tau ^{-1}\ge 0. \end{aligned}$$

The last inequalities are equivalent to the following:

$$\begin{aligned} B-D\le \alpha \le \frac{D_{p}\tau ^{-1}}{B-D}, \end{aligned}$$

i.e.,

$$\begin{aligned} B-D\le & \tau ^{-1}+d+D+D_{p}\le \frac{D_{p}\tau ^{-1}}{B-D}, \\ \left( B-D\right) \tau\le & 1+\left( d+D+D_{p}\right) \tau \le \frac{D_{p} }{B-D}. \end{aligned}$$

This is the condition when the parasites “kill” the hosts, since the only stationary point is (0, 0) , meaning total extinction.

Appendix 4: Proof of Theorem 4

From “Appendix 2”, we have that the dynamics of $x(t)=n(t)/K(t)$ are described by equation ${\dot{x}}(t)=\tau ^{-1}-a(t)x(t)+D_{p}x^{2}(t)$. Since the right-hand side is quadratic with two positive roots $x_{*}(t)<1$ and $x^{*}(t)>1$ (see “Appendix 2”), it can be presented in the form of the product of the roots. Describing the dynamics of x(t) as

$$\begin{aligned} {\dot{x}}(t)=D_{p}(x(t)-x^{*}(t))(x(t)-x_{*}(t)) \end{aligned}$$

directly leads to (49). From (58) in “Appendix 1”, we have that the dynamics of strategy frequencies (45) can be presented in the form

$$\begin{aligned} {\dot{q}}_{i}(t)=q_{i}(t)\left[ \left( b_{i}\dfrac{{\bar{d}}(t)}{{\bar{b}}(t)} -d_{i}\right) +\left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right) \left( \dfrac{ K(t)-n(t)}{n(t)\tau }-{\bar{d}}(t)\right) \right] , \end{aligned}$$

where the last bracketed term $\left( \dfrac{K(t)-n(t)}{n(t)\tau }-{\bar{d}} (t)\right)$ can be expressed in terms of $x(t)=n(t)/K(t)$, leading to (48).

Appendix 5: Proof of Theorem 5

Now, we should calculate the nullclines of the extended equation and their stability. The manifolds where the right-hand side of (51) equals zero will satisfy the equation

$$\begin{aligned} -D_{d}K^{2}+RK-D_{p}n=0 \end{aligned}$$

obtained from the bracketed term of (51). Here,

$$\begin{aligned} \Delta =R^{2}-4D_{d}D_{p}n \end{aligned}$$

and $R>2\sqrt{D_{d}D_{p}n}$ is the condition for the existence of two nullclines:

$$\begin{aligned} {\tilde{K}}_{1} & = \frac{R-\sqrt{R^{2}-4D_{d}D_{p}n}}{2D_{d}} \\ {\tilde{K}}_{2} & = \frac{R+\sqrt{R^{2}-4D_{d}D_{p}n}}{2D_{d}} \end{aligned}$$

where $K_{1}$ is an unstable extinction barrier and $K_{2}$ is the attracting nullcline. Since the additional factor $D_{d}K$ cannot affect Theorem 2, we can apply it and change coordinates in the system (50)–(51) from n, K to x, K. In effect, we obtain the system

$$\begin{aligned} {\dot{x}}(t) & = \frac{1}{\tau }-\left[ \frac{1}{\tau }+d+D_{p}+B\right] x(t)+D_{p}x^{2}(t), \end{aligned}$$

(64)

$$\begin{aligned} {\dot{K}}(t) & = K(t)\left[ R-D_{d}K(t)-D_{p}x(t)\right] . \end{aligned}$$

(65)

Then, Eq. (65) has the attracting nullcline ${\tilde{K}}=\dfrac{ R-D_{p}x}{D_{d}}$, and Eq. (64) converges to $x_{*}$. Combining them leads to equilibrium $x_{*}$ and ${\hat{K}}=\dfrac{ R-D_{p}x_{*}}{D_{d}}$, which is positive for $R>D_{p}x_{*}$.

Appendix 6: Proof of Theorem 6

Note that in the system (55)–(57), in addition to the strategy specific parasite mortalities $d_{i}$, we have additional mortalities caused by the death of the host equal to $D-D_{p}-D_{d}K(t)$. We can incorporate them into aggregated mortalities $d_{i}^{A}(K(t))=d_{i}+D+D_{p}+D_{d}K(t)$ (and thus, ${\bar{d}}^{A}(t,K(t))={\bar{d}} (t)+D+D_{p}+D_{d}K(t)$). Then, analogously to (58) from “Appendix 1” and (48), the frequency dynamics can be presented as

$$\begin{aligned} {\dot{q}}_{i}(t) & = q_{i}(t)\left[ \left( b_{i}-{\bar{b}}(t)\right) \left( \dfrac{K(t)-n(t)}{n(t)\tau {\bar{b}}(t)}\right) \right. \\&\quad \left. -\left( d_{i}^{A}(K(t))-{\bar{d}} ^{A}(t,K(t))\right) \right] = \\ & = q_{i}(t)\left[ \left( b_{i}\dfrac{{\bar{d}}^{A}(t,K(t))}{{\bar{b}}(t)} -d_{i}^{A}(K(t))\right) \right. \\&\quad \left. +\left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right) \left( \dfrac{K(t)}{n(t)\tau }-\tau ^{-1}-{\bar{d}}^{A}(t,K(t))\right) \right] . \end{aligned}$$

On the nullcline ${\tilde{n}}(t,K(t))$ (52), where the bracketed term of the r.h.s of the parasite population size equation (56) is zero, we have that ${\tilde{x}}(t,K(t))={\tilde{n}}(t,K(t))/K(t)$, and in effect, we obtain

$$\begin{aligned} 1/{\tilde{x}}(t,K(t))\tau & = \dfrac{K(t)}{{\tilde{n}}(t,K(t))\tau }\\& =\tau ^{-1}+ {\bar{d}}(t) +D+D_{p}+D_{d}K(t) \\& =\tau ^{-1}+{\bar{d}}^{A}(t,K(t)), \end{aligned}$$

and this factor is an increasing function of K. Then, the frequency dynamics can be presented as

$$\begin{aligned} {\dot{q}}_{i}(t) & = q_{i}(t)\left[ \left( b_{i}\dfrac{{\bar{d}}^{A}(t,K(t))}{{\bar{b}} (t)}-d_{i}^{A}(K(t))\right) \right. \\&\left. +\left( \dfrac{b_{i}}{{\bar{b}}(t)}-1\right) \left( \dfrac{1}{x(t)\tau }-\dfrac{1}{{\tilde{x}}(t,K(t))\tau }\right) \right] \end{aligned}$$

Recall the proof of Theorem 2. Therefore, we can replace Eq. (56) by the equation

$$\begin{aligned} {\dot{x}}(t)=-D_{p}(x(t)-x^{*}(t))(x(t)-x_{*}(t)) \end{aligned}$$

as in Theorems 4 and 5. Then, the factor $\dfrac{1}{ x(t)\tau }$ will be attracted by $\dfrac{1}{x_{*}(t)\tau }$. Similar to Theorem 5, Eq. (57) can be denoted in terms of x(t) as

$$\begin{aligned} {\dot{K}}(t)=K(t)\left[ B-D-D_{d}K(t)-D_{p}x(t)\right] . \end{aligned}$$

Thus, the density subsystem (56)–(57) is attracted by $x_{*}(t)$ and ${\hat{K}}(t)=\dfrac{R-D_{p}x_{*}(t)}{D_{d}}$, and $R\le D_{p}x_{*}(t)$ is the condition for evolutionary suicide.

Now, fix the value of q(t) and focus on the subspace n, K (which can be presented as x, K). For each particular point q, we have a specific value of ${\bar{d}}(q)$. From Theorem 5, we know that for every fixed value of d, a unique stable equilibrium $x_{*}$ exists, which determines the respective stable value of ${\hat{K}}$. Since this is a stationary point, it should be the intersection of the attracting nullcline ${\tilde{n}}(t,K)$ and the attracting nullcline ${\tilde{K}}_{2}$ (54) from Theorem 5. Therefore, at this point, we have that $x_{*}={\tilde{x}}(t,{\hat{K}})$ and in effect

$$\begin{aligned} \left( \dfrac{1}{x(t)\tau }-\dfrac{1}{{\tilde{x}}(t,K(t))\tau }\right) =0. \end{aligned}$$

(66)

For every value of q from the strategy simplex, we can define point $x_{*}$, and for that point, we can define the respective ${\hat{K}}$. Both $x_{*}$ and ${\hat{K}}$ constitute, respectively, the attracting surfaces for system (56, 57). Since ${\hat{K}}$ depends only on $x_{*}$, which in turn depends on the value of ${\bar{d}}$ determined by the strategic composition of the population q(t), these attracting surfaces can be described as

$$\begin{aligned} \breve{x}_{*}(q) & = \frac{a(q)-\sqrt{a(q)^{2}-4D_{p}\tau ^{-1}}}{2D_{p}} ,\\ \quad {\hbox {where}}\,\, a(q) & = \tau ^{-1}+{\bar{d}}(q)+D_{p}+B, \\ \breve{K}(q) & = \dfrac{R-D_{p}\breve{x}_{*}(q)}{D_{d}}. \end{aligned}$$

On the surface $\left( \breve{x}_{*}(q),\breve{K}(q)\right)$, condition (66) is always satisfied, leading to the pure nest site lottery mechanism operating there.