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.