Persistence and size of seasonal populations on a consumer–resource relationship depends on the allocation strategy toward life-history functions

The long-term population dynamics (i.e., extinction or persistence in an equilibrium value) of the system (2) is mainly dependent on the constant allocation of energy between life-history functions. In the case of population persistence, there is a unique long-term allocation strategy towards reproductive functions that maximizes the population abundance and minimizes individual consumption. In addition, this strategy is dependent on the parameters associated with both fertility ((kappa ) and ({mathscr {B}}_0)) and survival ((xi _{c}) and (e_{1/2})) costs.

Preliminary results

To investigate long-term dynamic patterns, we will relate the energy states of reproduction and maintenance, resource density and population abundance at the end of reproductive seasons, namely in the time sequence ({ntau ^+}_{nge 0}). In the system (4) the relationships between the state variables are the same for each (tau ) unit of time. Then, there is a transformation that relates to the vector ( (R,E_ {r}, E_ {m},P) ((n + 1)tau ^+) ) with ( (R,E_ {r}, E_ {m}, P) (n tau ^+) ) such that ( (n + 1) tau ^+ – ntau ^+ = tau ). This relationship is determined by the following discretization (or stroboscopic map) of the impulsive differential system (4):

and (R(ntau ^+)=K), where the function (Phi ) defined by (5) is evaluated at (t=(n+1)tau ) and extended to (p=0):

for any (nge 0). Indeed, in non-reproductive seasons the consumer–resource dynamics are described by the continuous component of the system (4), which can be solved. Directly we have (P(t)=P(ntau ^+)e^{-lambda (t-ntau )}) for any (tin (ntau ,(n+1)tau ]). In addition, the per capita consumption rate (1) can be expressed in terms of ( R'(t)). Therefore, reproductive and maintenance energy rates are determined by

Integrating these functions in the interval ( (n tau , t] ), we obtain

Evaluating Eqs. (8)–(9) at ( t = (n + 1)tau ) (end of the non-reproductive season), we have

with (Phi (n,P(ntau ^+))= Phi ((n+1)tau ,R((n+1)tau ),P(ntau ^+))) and (P((n+1)tau )=P(ntau ^+)e^{-lambda tau }) for any (nge 0). In addition, evaluating the discrete component of system (4) at (t=(n+1)tau ), we obtain

Finally, substituting the Eqs. (10)–(11) into the Esq. (12)–(14) we obtain the discretization given by Eq. (6).

In order to obtain the equilibrium points of system (6), we can solve the following equations:

where (e_{r}:=lim _{nrightarrow +infty }{E_{r}(ntau ^+)}), (e_{m}:=lim _{nrightarrow +infty }{E_{m}(ntau ^+)}), (rho :=lim _{nrightarrow +infty }{P(ntau ^+)}) and then, (Phi (rho ):=lim _{nrightarrow +infty }{Phi (n,P(ntau ^+))}). From the first equation, we have (e_{r}=(1-kappa gamma )alpha Phi (P)/kappa gamma ) and then (e_{m}=kappa {mathscr {B}}_0 (1-alpha )xi _{c}/alpha ) for (Pin {0,rho }) such that (Phi (0)=R_{max}(e^{lambda tau }-1)/lambda ) and

Therefore, assuming (lambda =0), we have (Gamma (rho )=K-{mathscr {A}}rho ) (from equations (7) and (15)) where (Gamma :=lim _{nrightarrow +infty }{R((n+1)tau )}), (rho ) is the solution of

if, and only if,

Long-term population dynamics

From the discretization (6), there are two dynamic behaviors for the long-term population abundance: extinction (see Fig. 2a) and persistence (see Fig. 2b).

The differentiation of these behaviors strongly depends on the individual consumption of resources defined throughout each non-reproductive season by

At the end of each non-reproductive season, namely at the time ( t = (n + 1) tau ), the individual consumption is given by

where (R((n+1)tau )) is the non-consumed resource density by the population, the amount that is obtained from the implicit solution of the resource density equation in the continuous component of the system (4). Thus, projecting the individual consumption of the resource into the long term, and taking (Gamma (rho )=(K-{mathscr {A}}rho +lambda {mathscr {I}})e^{-lambda tau }), the expression (17) assumes the form

where ({mathscr {I}}:=lim _{nrightarrow +infty }{int _{ntau }^{(n+1)tau }{R(s)e^{lambda (s-ntau )}ds}}). On the one hand, the individual consumption ( C_{I}^{infty } ) is composed of a basis amount corresponding to the term ({mathscr {A}} ) and an amount resulting from the equal division of a resource not consumed by individuals dying during the non-reproductive season, ( K(e^{lambda tau } -1) – lambda {mathscr {I}}). Furthermore, when the population experiences a reduced mortality during the non-reproductive season (i.e., ( lambda approx 0) ), the individual consumption is ( C_{I}^{infty } (rho ) approx {mathscr {A}} ). On the other hand, whether the population abundance is low, the per capita resource is high, implying an individual consumption close to (R_ {max} (e^{lambda tau } -1) / lambda ) (equivalent to taking the limit of ( C_{I}((n+1)tau )) as ( P(ntau ^+)rightarrow 0 )). Certainly, this quantity does not represent the effective individual consumption, but rather establishes an upper limit for this and therefore represents a value of non-persistence. Thus, behaviors related to the equilibrium solutions of the discrete system (6) can be differentiated by the threshold value

In particular, when the mortality of the population is low, the long-term abundance is described by Eq. (16) and the threshold value assumes the following form ({mathscr {U}}=R_{max}tau /{mathscr {A}}). Thus, we conclude that the persistence of the population is established when ({mathscr {U}}>1) and extinction when ({mathscr {U}}le 1).

Finally, we can see that the stabilization of population size in the long term is in response to a dense-dependent behaviour where the per capita growth rate in the long term is (r_{infty }:={alpha {mathscr {A}}/kappa }{mathscr {S}}(e_m)) equivalent to mortality fraction (mu ), where (e_{m}=kappa {mathscr {B}}_{0}(1-alpha )xi _{c}/alpha ) is the equilibrium value of energy maintenance after the reproductive season. In addition, the derivative of r with respect to P is

where (dPhi /d P<0), (dr/d P<0) and (dr/dPhi >0) are obtained, which explains the expected dense-dependency.

Then, we can derive the following conclusion.

Theorem 1

We consider the threshold value ({mathscr {U}}=R_{max}tau /{mathscr {A}}).

1. 1.

   If ({mathscr {U}}le 1) then the long-term population behavior is extinction.

2. 2.

   If ({mathscr {U}}>1) then the long-term population behavior is persistence.

Proof

We divide the proof into two cases: (kappa gamma =1) and (0<kappa gamma <1). In the first case we define the functions

Therefore, the discrete system defined by equations (e_{m}^{k+1}=G(e_{m}^{k},p^{k})) and (p^{k+1}=H(e_{m}^{k},p^{k})) with (kge 0) has two equilibrium solutions: ({mathscr {E}}(P)=(E_{m},P)) where (E_{m}={mathscr {B}}_0 (1-alpha )xi _{c}/alpha gamma ) and (Pin {0,rho }). Thus,

with (Phi ‘(p)=-r_{0}Phi (p)/[r_{0}p+Gamma (p)]) where ((,)’) means the derivative with respect to p. Then, the matrix of linearization around the equilibrium point ({mathscr {E}}(P)) is given by the Jacobian matrix in ({mathscr {E}}(P)),

Therefore, ({mathscr {E}}(0)) is uniformly asymptotically stable and ({mathscr {E}}(rho )) is locally asymptotically stable.

In the second case, (0<kappa gamma <1), we define the functions

Then, the discrete system is defined by equations (e_{r}^{k+1}=F(e_{r}^{k},e_{m}^{k},p^{k})), (e_{m}^{k+1}=G(e_{r}^{k},e_{m}^{k},p^{k})) and (p^{k+1}=H(e_{r}^{k},e_{m}^{k},p^{k})) with (kge 0) which has two equilibrium solutions: ({mathscr {E}}(P)=(E_{r}(P),E_{m},P)) where (E_{r}(P)=(1-kappa gamma )alpha Phi (P)/kappa gamma ), (E_{m}=kappa {mathscr {B}}_0 (1-alpha )xi _{c}/alpha ) and (Pin {0,rho }). Then, (partial F/partial e_r=(1-kappa gamma )), (partial F/partial e_m=0), (partial F/partial p=(1-kappa gamma )alpha Phi ‘(p)), (partial H/partial e_{r}=gamma p G / (e_{1/2}+G)),

and

The analysis of stability for the equilibrium solution ({mathscr {E}}(0)) is equivalent to that carried out in the first case. Thus, ({mathscr {E}}(0)) is uniformly asymptotically stable.

To analyze the stability of the equilibrium point ( {mathscr {E}}: = {mathscr {E}}(rho ) ), we consider the characteristic polynomial of the linearization matrix around this point, (tilde{F}(x)=x^3+ax^2+bx+c=0), where

From the sign of the derivatives (a,, c<0) and (b>0) are obtained. Thus, the necessary conditions of Jury’s criteria: (i) ({tilde{F}}(1)>0), (ii) ({tilde{F}}(-1)<0) and (iii) (|c|<1), are met. Indeed, ({tilde{F}}(1)=kappa gamma A_{1}A_{3}[1+(1-kappa gamma )A_{2}A_{3}]), ({tilde{F}}(-1)=-(1+|a|+b+|c|)) and (|c|=(1-kappa gamma )(1-A_3)(1+kappa gamma A_1 A_2 A_3)) where (A_{1}=mu r_{0}rho /(r_0rho +K-{mathscr {A}}rho )), (A_2=e_{1/2}/(e_{1/2}+E_{m})) and (A_{3}=alpha {mathscr {A}}/(kappa {mathscr {B}}_0+alpha {mathscr {A}})) with (0<A_{i}<1) for (iin {1,2,3}). Note that (|c|<(1-A_3)[1-(kappa gamma )^2]<1). In addition, the sufficient condition (1-c^2>|ac-b|) is equivalent to (c^2-1<ac-b<1-c^2). From (i), we have that (1-c^2-(ac-b)+(a-bc)>0) and (1-c^2-(ac-b)-(a-bc)>0). Adding these expressions, (ac-b<1-c^2) is obtained. Whereas for analyzing the inequality (c^2-1<ac-b) we consider the expression

Note that in Eq. (19), ( kappa gamma ) is the amount that provides the greatest variability to this expression. Then, we define the function

for any (xin [0,1]). Now, we will show that (0<g(x)<1) for any (xin [0,1]). The function g is a fourth degree polynomial (g(x)=B_{4}x^4+B_{3}x^3+B_{2}x^2+B_{1}x+B_0) with coefficients

where the values of (B_3) and (B_1) depend on the following three disjoint regions (see Fig. 3):

Some important properties of the function g are: (g(0)=B_0in (0,1)), (g(1)=(1-A_1)(1-A_3)>0), (g(0)-g(1)=A_1(1-A_3)+A_3(1-A_1A_2A_3)>0) which implies that (0<g(1)<g(0)<1). Also, (g'(0)=B_1) and (g'(1)=-{(1-A_3)(2A_1+A_3)+A_3(1-A_1 A_2)^2+A_1 A_2 A_3^{2}(A_1+3-A_3)}<0) (see Fig. 4).

Therefore, by the method of Descartes, we have the five admissible cases which are presented in Table 1. On the one hand, in the cases (C_1), (C_3), (C_4) and (C_5), (g’) has there is only one positive real root ({overline{x}}={overline{x}}(A_1,A_2,A_3)) where ((, )’) means the derivative with respect to x. Note that ({overline{x}}>1), because otherwise there would be at least one additional zero of (g’) on the interval (0, 1) for the basic properties are met. Then, the function g is a positive and decreasing function for any (xin [0,1]) with maximum value (g(0)=B_0<1). On the other hand, in case (C_2), (g’) has one positive real root ({overline{x}}in (0,1)) at which the polynomial g has a local maximum. Taking (g({overline{x}})) to define a function (f(A_1,A_2,A_3)) for ((A_1, A_2, A_3)in {mathscr {R}}_3), we have that (0<f(A_1,A_2,A_3)<1) if and only if ((A_1,A_2,A_3)in {mathscr {R}}_3) (using the application Mathematica). Thus, in all cases, (0<g(x)<1) for any (xin [0,1]), which implies (c^2-ac+b<1) and hence (c^2-1<ac-b). Therefore, ({mathscr {E}}(rho )) is locally asymptotically stable. (square )

Optimal strategy

The validity of the condition ( {mathscr {U}}> 1 ) depending on ( alpha ) establishes a range of allocation strategies ( (alpha _m, alpha _M) ) in the which the persistence of the population is guaranteed (see Fig. 5). Thus, ( alpha in (alpha _{m}, alpha _{M}) ) if and only if ( {mathscr {U}}> 1 ), where (alpha _{m}) and (alpha _{M}) are the limits of this range such that ( {mathscr {U}} = 1 ) is obtained.

In this persistence range we find a unique allocation strategy described by

which maximizes the population abundance and minimizes the individual consumption in the long term (see Fig. 6). This allocation strategy satisfies ( rho ‘(alpha _ {op}) = 0) if, and only if, also satisfies ({mathscr {A}}'(alpha _ {op}) = 0 ) where ( (,)’ ) means the derivative with respect to (alpha ) and

In addition, the strategy ( alpha _ {op}) is dependent on the parameters associated with both the costs of fertility ((kappa ) and ({mathscr {B}}_0)) and survival ((xi _{c}) and (e_{1/2})), so that if the fertility costs (Delta :=kappa {mathscr {B}}_0) are greater than the maintenance ratio (delta :=e_{1/2}/xi _{c}), the allocation is to favor reproduction, i.e., (alpha _{op}>0.5) (see Fig. 6a). Otherwise, the allocation is to favor maintenance, i.e., (1-alpha _{op}ge 0.5) (see Fig. 6b).

Source: Ecology - nature.com