Abstract
Many plant populations have persistent seed banks, which consist of viable seeds that remain dormant in the soil for many years. Seed banks are important for plant population dynamics because they buffer against environmental perturbations and reduce the probability of extinction. Viability of the seeds in the seed bank can depend on the seed’s age, hence it is important to keep track of the age distribution of seeds in the seed bank. In this paper we construct a general density-dependent plant-seed bank model where the seed bank is age-structured. We consider density dependence in both seedling establishment and seed production, since previous work has highlighted that overcrowding can suppress both of these processes. Under certain assumptions on the density dependence, we prove that there is a globally stable equilibrium population vector which is independent of the initial state. We derive an analytical formula for the equilibrium population using methods from feedback control theory. We apply these results to a model for the plant species Cirsium palustre and its seed bank.
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Acknowledgments
The authors would like to thank professors Glenn Ledder and Steve Dunbar for their helpful comments about the mathematical model during the final stages of this work. The authors would also like to thank Associate Editor Sebastian Schreiber and the two anonymous reviewers for their constructive suggestions that greatly improved the quality of the manuscript.
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Appendices
Appendix A
Proof of Theorem 3.2
Without loss of generality, we can assume that \(n_0\) is in \(K_1 \setminus \{0\}\). If it is not, then \(s_i \ne 0\) for some \(i = 1, 2, \ldots N\), which would imply that \(n_1 \in K_1 \setminus \{0\}.\)
To prove part (1) of Theorem 3.2, since \((\tilde{c}_{1}^T(\tilde{I}-\tilde{A}_{1})^{-1}\tilde{b})^{-1} > g_0 = \sup _{y>0} g(y)\) and \(h(y) \le y\),
for some \(m < p_1\). By induction
Since \(p_1 = (\tilde{c}_{1}^T(\tilde{I}-\tilde{A}_{1})^{-1}\tilde{b})^{-1}\) is the stability radius of \((\tilde{A}_{1}, \tilde{b}, \tilde{c}_{1}^T)\), we have that \(r(\tilde{A}_{1} + m\tilde{b}\tilde{c}_{1}^T)<1\). Thus
The \((\epsilon , \delta )\) conclusion follows from the boundedness of \(\tilde{A}_{1} + m\tilde{b}\tilde{c}_{1}^T\).
For (2), with the triple \((p_1, p_2, y^*) \in (0,g_0)\times (0,1)\times (0,\infty )\) satisfying (3.10) define the functional
It is straightforward to verify that
Applying \(\tilde{w}_{p_2}^T\) to (2.7),
If \(\tilde{y}_t \le y^*\) and \(c^Tn_t \le c^Tn^*\), then, since both \(f\) and \(h\) are increasing, concave down with \(f(0) = h(0) = 0\), we have that \(f(\tilde{y}_t) \ge p_1y_t\) and \(h(c^Tn_t) \ge p_2c^Tn_t\), so (6.3) implies that
If \(\tilde{y}_t \le y^*\) and \(c^Tn_t \ge c^Tn^*\), then \(f(\tilde{y}_t) \ge p_1\tilde{y}_t\) and \(h(c^Tn_t) \ge p_2c^Tn^*\), so (6.3) implies that
If \(\tilde{y}_t \ge y^*\), then \(f(\tilde{y}_t) \ge p_1y^*\), so (6.3) implies that
Hence (6.4), (6.5) and (6.6) imply that
By Holder’s inequality
Using again that either \(h(c^Tn_t) \ge p_2c^Tn_t\) or \(h(c^Tn_t) \ge p_2c^Tn^* = \frac{p_1p_2y^*}{p_e}\), it follows from (6.8) that
Finally, since \(\tilde{n}_0\) is a positive vector in \(X_1\otimes X_2\),
Thus,
Similarly, \(p_2p_1\alpha _1\tilde{w}_{p_2}^T\tilde{b}c^Tn^*\), and \(\tilde{w}_{p_2}^T\tilde{b}p_1y^*\) are positive, so \(\tilde{y}_t\) is bounded away from zero for all \(t>0\). Also, using condition (E3) and the fact that \(n_0 \in K_1 \setminus \{0\}\), \(c^Tn_t\) is bounded away from zero for all \(t>0\), by a similar argument. Thus, since \(f\) and \(h\) are increasing and concave down (see Fig. 6), from the secant slopes
we can find \(m_1 < p_1\) and \(m_2 < p_2\) such that for all \(t \ge 0\),
We can easily verify from (3.8) that \(\tilde{n}^* = \tilde{A}_{p_2}\tilde{n}^* + p_1\tilde{b}\tilde{c}_{p_2}^T\tilde{n}^*=\tilde{A}\tilde{n}^* + \tilde{b}f(y^*)\) by construction. Thus
Since \(\tilde{A}\) is nonlinear, the variation of parameters formula becomes
Since the \(X_1 \rightarrow X_1\) and \(X_2 \rightarrow X_1\) components of \(\tilde{A}\) (\([A \; \; \emptyset ]\)) are linear, and \(\tilde{b} = [b \; \; 0]^T\) we have that the \(X_1\) component of \(\tilde{n}_t - \tilde{n}^*\) satisfies
Multiplying (6.15) on the left by \(\tilde{c}_{p_2}^T\), we have
Taking absolute values and using positivity gives us that
Using (6.12),
Summing from \(t = 0\) to \(M\), where \(M\) is large, we have
Since \(r(A) < 1\) the first term in (6.18) converges as \(M \rightarrow \infty \). If we rearrange the second sum and use the fact that the system is positive, we have
Adding more terms and changing indices
The \(X_2\) component of \(\tilde{n}_t\) satisfies
where \(\Gamma _1 := [\gamma _1 \; \; 0\, \cdots \,0]^T\). Since
it follows that
Since \(\tilde{n}^*\) was is a fixed point of our system, if we insert \([n^* \; \; s^*]^T\) for \([n _0 \; \; s_0]^T\) in (6.20) we obtain
Thus, subtracting (6.22) from (6.21), and multiplying by \(\alpha ^T\) on the left gives us
Using (6.12), and the positivity of the system, we have that
Putting
and
back into (6.24) we obtain
Summing from \(t = 0\) to \(M\), for \(M\) large, and rearranging we have
Adding more terms and changing indices, we have
Using (6.12) again,
Using the triangle inequality, as well as (6.12) again,
Since \(\sum _{j=0}^{\infty }S^j= (I - S)^{-1}\) and \(\sum _{k=0}^{\infty }A^k= (I - A)^{-1}\) we have
The first two terms of (6.25) converge as \(M \rightarrow \infty \), since \(r(A) < 1\) and \(\gamma _i < 1\) for all \(i\). Define
Adding the (6.19) and (6.25) together, we obtain
Since \(m_1 < p_1\) and \(m_2 < p_2\), and using (3.7) there exists an \(m < 1\) such that
Hence
which implies that
This bound is independent of \(M\). Therefore the sequence
so
which, by the continuity of \(h\), implies that
By (6.13), (6.28) and assumptions (E1) and (E4) we therefore have that
as sought. The \((\epsilon , \delta )\) conclusion follows from Holder’s inequality and assumption \((E1)\). \(\square \)
Appendix B
Proof of Theorem 3.3
The proof of (1) is identical to the proof of (1) in Theorem 3.2.
For (2) note that if there exists a solution \((p_1, p_2, y^*)\) of (3.10) in \((0,g_0)\times (\exp (-2),1)\times (0,\infty )\) and \(m > 0\) such that \(\tilde{y}_t > m\) and \(c^Tn_t \ge m\) for all \(t \in \mathbb{N }\), then \(h_R\) is sector bounded as in (3.11) (Fig. 2). This follows from the fact that
Thus \(h_R\) has \(\mathrm exp (-2)\) as its maximum negative slope. If \(\tilde{y}_t\) and \(c^Tn_t\) are uniformly bounded away from \(0\), \(h_R\) satisfies
for some \(m_2 < p_2\). To see that there exists \(m > 0\) such that \(\tilde{y}_t\ge m\) and \(c^Tn_t \ge m\) for all \(t \in \mathbb{N }\), we note that if \(\tilde{y}_t \le y^*\) and \(c^Tn_t \le c^Tn^*\) or \(\tilde{y}_t \ge y^*\) the lower bound follows as in Theorem 3.2. If \(\tilde{y}_t \le y^*\) and \(c^Tn_t \ge c^Tn^*\) we need to show that the solution \(\{\tilde{n}_t\}_{t=0}^{\infty }\) is bounded above. Noting that \(f(y) \le f(y^*) + m_1y\) for some \(m_1 < p_1\) and \(y \ge 0\) and \(h_R(y) \le c_m\mathrm exp (-1)\) for all \(y \ge 0\), it follows that
where
and \(r(\hat{A}) <1\). Thus \(\tilde{c}_1^T\tilde{n}_t \le M\) for some \(M < \infty \), which implies that \(c^Tn_t \le M/\alpha _1\) for all \(t \ge 0\). Thus, if \(\tilde{y}_t \le y^*\) and \(c^Tn_t \ge c^Tn^*\) we have that \(f(\tilde{y}_t) \ge f(y^*)\) and \(h_R(c^Tn_t) > \mathrm min \{h_R(c^Tn^*), h_R(M/\alpha _1)\} > 0\). Letting \(\tilde{w}_{p_2}^T\) be defined as in Theorem 3.2,
thus \(\tilde{y}_t\), and similarly \(c^Tn_t\), are bounded from below as in Theorem 3.2. The remainder of the proof for (2) is the same as in Theorem 3.2.
For part (3) note that
so we cannot sector-bound \(h_R\) as we did in (2) of this theorem. The linearization about \(\tilde{n}^*\) yields
Thus if \(r(\tilde{A}_{(1 +\ln (p_2))p_2} + f^{\prime }(y^*) \tilde{b}\tilde{c}_{(1 +\ln (p_2))p_2}^T) < 1\) then \(\tilde{n}^*\) is asymptotically stable, as sought. \(\square \)
Appendix C
Proof of Corollary 3.1
To prove (1) we need to show that \((\tilde{A},\tilde{b},\tilde{c})\) satisfies hypotheses (A1), (A2) and (A3) in Rebarber et al. (2012). We verified that (A1) and (A2) are met by \(\tilde{A}\) and \(\tilde{b}\) in the proof of Theorem 3.1. To prove that (A3) is met by \(\tilde{c}^T\), note that (3.16) implies that \(c^Tn \ge 0\) for all \(n \in X_1\). This, coupled with (E4) implies that \(\tilde{c}^T\tilde{n} \ge 0\) for all \(\tilde{n} \in X_1 \otimes X_2\), proving (1).
To prove (2) and (3) note that (E3) is only used in the proof of Theorem 3.2 (2) and Theorem 3.3 (2), i.e. when \(\tilde{n}^*\) is positive and globally stable. Also, the only place where we needed to use (E3) in the proofs of Theorems 3.2 and 3.3 is where we assert that there exists an \(m > 0\) such that \(\tilde{y_t},c^Tn_t \ge m\) for all \(t \ge 0\). To prove this in the case where \(h\) is continuous, increasing, concave down, with \(h(0) = 0\), we introduce a new IPM system which is “close”’ to the original system (2.7). For \(\epsilon >0\), let \(I_{\epsilon }:=\{x \in [L,U]| c(x) > \epsilon \}, X_{1,\epsilon }:= L^1(I_{\epsilon })\) and \(A_{\epsilon }:X_{1,\epsilon } \rightarrow X_{1,\epsilon }\) be such that
with \(b_{\epsilon } = b|_{I_\epsilon }\) and
It follows that \(r(A_{\epsilon }) \le r(A) < 1\), so \(A_{\epsilon }\) satisfies (E1). It’s straightforward that, for sufficiently small \(\epsilon \), \(b_{\epsilon }\) satisfies (E2) and that \(c_\epsilon ^T\) satisfies (E3) with \(c_{\min } = \epsilon \). Let
It follows that \(\lim _{\epsilon \rightarrow 0}p_e(\epsilon ) = p_e\). Since in Theorem 3.2 (2) the system of Eqs. (3.10) has a solution \((p_1,p_2,y^*) \in (0,g_0)\times (0,1)\times (0,\infty )\), we can choose \(\epsilon >0\) such that the system of equations
has a solution \((p_1(\epsilon ),p_2(\epsilon ),y^*(\epsilon )) \in (0,g_0)\times (0,1)\times (0,\infty )\). Let \([n(\epsilon )_t \; \; s(\epsilon )_t]^T \subset X_{\epsilon }:= X_{1,\epsilon }\otimes X_2\) solve
Since \((p_1(\epsilon ),p_2(\epsilon ),y^*(\epsilon )) \in (0,g_0)\times (0,1)\times (0,\infty )\) we have, by the proof of Theorem 3.2, the existence of an \(m >0\) such that \(\tilde{y}(\epsilon ), c_\epsilon ^Tn(\epsilon )_t \ge m\) for all \(t \ge 0\). By the monotonicity of \(f\) and \(h\) and the positivity of \((A,b,c)\), \(\{\alpha _j\}_{j=1}^{N+1}\) and \(\{\gamma _j\}_{j=1}^{N+1}\) we have \(\tilde{y}_t \ge \tilde{y}(\epsilon )_t\) and \(c^Tn_t \ge c_\epsilon ^Tn(\epsilon )_t\) for all \(t \ge 0\). Thus there exists an \(m >0\) such that \(\tilde{y}, c^Tn_t \ge m\) for all \(t \ge 0\) and (2) is proved.
For (3) \(h(y)\) is equal to \(h_R(y) = y e^{-y/c_m}\), which is not monotone once \(y\) becomes larger than \(c_m\). Thus we cannot bound \(\tilde{y}_t\) and \(c^Tn_t\) from below by \(\tilde{y}(\epsilon )_t\) and \(c(\epsilon )^Tn(\epsilon )_t\) for all \(t \ge 0\), unless \(c^Tn_t\) does not exceed \(c_m\) for all \(t \ge 0\).
In the proof of Theorem 3.3, we showed that there exists an \(M >0\) (which depends on \(\tilde{n}_0\)) such that \(\tilde{c}^T_1\tilde{n}_t \le M\) for all \(t \ge 0\), which implies that \(c^Tn_t \le M/\alpha _1\) for all \(t \ge 0\). If \(M/\alpha _1 \le c_m\) we can use the same arguments as in (2), due to the monotonicity of \(h_R(y)\) for \(y \le c_m\). If \(M/\alpha _1 > c_m\) we will construct a seed production function \(\underline{h}\) that is continuous, increasing, concave down, with \(\underline{h}(0) = 0, \underline{h}^{\prime }(0) = 1\) such that \(\underline{h}(y) \le h_R(y)\) for all \(y \in [0,M/\alpha _1]\). The population in a model with \(\underline{h}\) instead of \(h_R\) will be smaller than \(\tilde{n}_t\) for all \(t \ge 0\). If this smaller population has a nonzero globally stable equilibrium population than we will have our desired lower bound.
Define the function
which is continuous, increasing, concave down, with \(\underline{h}(0) = h_R(0) = 0, \underline{h}^{\prime }(0) = h_R^{\prime }(0) = 1\) and \(\underline{h}(M/\alpha _1) = h_R(M/\alpha _1) = M/\alpha _1 e^{-\frac{M}{\alpha _1 c_m}}\). Since \(c_m < M/\alpha _1\) implies that
we have that \(\underline{h}(y) \le h_R(y)\) for all \(y \in [0,M/\alpha _1]\) (see Fig. 7). Let \(\tilde{\underline{n}}_t = [\underline{n}_t \; \; \underline{s}_t]^T \subset X\) solve
It follows that, if \(\tilde{\underline{n}}_0 = \tilde{n}_0\), we have that \(\tilde{\underline{n}}_t \le \tilde{n}_t\), and thus \(c^T\underline{n}_t \le c^Tn_t\), for all \(t \ge 0\). Since in Theorem 3.3 (2) the sytem of Eqs. (3.10) has a solution \((p_1,p_2,y^*) \in (0,g_0)\times (e^{-2},1)\times (0,\infty )\) it follows that the system of equations
has a solution \((\underline{p}_1,\underline{p}_2,\underline{y}^*) \in (0,g_0)\times (0,1)\times (0,\infty )\) (see Fig. 7). This implies, from the above proof of (2), that \(\tilde{\underline{n}}_t\) converges to a positive, globally stable equilibrium population \(\tilde{\underline{n}}^*\). Since \(\tilde{n}_t \ge \tilde{\underline{n}}_t\) for all \(t \ge 0\) we have the desired positive lower bound \(m\) for \(\tilde{y}_t\) and \(c^Tn_t\), proving (3). \(\square \)
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Eager, E.A., Rebarber, R. & Tenhumberg, B. Global asymptotic stability of plant-seed bank models. J. Math. Biol. 69, 1–37 (2014). https://doi.org/10.1007/s00285-013-0689-z
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DOI: https://doi.org/10.1007/s00285-013-0689-z
Keywords
- Structured population model
- Age-structured seed bank
- Density dependence
- Global asymptotic stability
- Contest competition
- Scramble competition