Dynamics of Planar Systems That Model Stage-Structured Populations
We study a general discrete planar system for modeling stage-structured populations. Our results include conditions for the global convergence of orbits to zero (extinction) when the parameters (vital rates) are time and density dependent. When the parameters are periodic we obtain weaker conditions for extinction. We also study a rational special case of the system for Beverton-Holt type interactions and show that the persistence equilibrium (in the positive quadrant) may be globally attracting even in the presence of interstage competition. However, we determine that with a sufficiently high level of competition, the persistence equilibrium becomes unstable (a saddle point) and the system exhibits period two oscillations.
Stage-structured models of single-species populations with lowest dimension in discrete time are expressed as planar systems of difference equations. For a general expression of these models, consider systemfrom  in which and are numbers (or densities) of juveniles and adults, respectively, remaining after (juvenile) periods. The vital rates and (survival and inherent low density fertility) as well as the competition coefficients and in (1a) and (1b) may be density dependent; that is, they may depend on and and also explicitly on time; that is, the system may be nonautonomous. Early examples of matrix models used in species populations dynamics can be found in [2–5] and their comprehensive treatment is provided in .
Under certain constraints on the various functions, including periodic vital rates and competition coefficients having the same common period , sufficient conditions for global convergence to zero (extinction) as well as the existence of periodic orbits for (1a) and (1b) are established in . If is the mean fertility rate (the mean value of above), then it is also shown that orbits of period appear when exceeds a critical value , while global convergence to 0 or extinction occurs when . On the other hand, conditions under which the species survives (i.e. permanence) were studied in [7, 8].
In this paper, we study the following abstraction of the matrix model (1a) and (1b): where for each time period the functions are bounded on the compact sets in . This feature allows for to be a fixed point of the system and it is true if, for example, , , are continuous functions for every . Under biological constraints on the parameters, we may think of and as in (1a) and (1b).
System (2a) and (2b) includes typical stage-structured models in the literature. For instance, the tadpole-adult model for the green tree frog Hyla cinerea population that is proposed in  may be expressed as
This is a system of type (2a) and (2b) with Beverton-Holt type functions and . Competition in (3a) and (3b) occurs separately among juveniles and adults but not between the two classes, as they feed on separate resources; thus and do not depend on both juvenile and adult numbers and is independent of both numbers. Two cases are analyzed in : (i) continuous breeding with constant so that (3a) and (3b) is autonomous and (ii) seasonal breeding where is periodic. In addition to considering extinction and survival in the autonomous case, it is shown that seasonal breeding may be deleterious (relative to continuous breeding) for populations with high birth rates, but it can be beneficial with low birth rates.
The numbers denote the harvest rates of juveniles and adults, respectively. The stock-recruitment function may be compensatory (e.g., Beverton-Holt ) or overcompensatory (e.g., Ricker ). Compensatory recruitment is used in populations where recruitment increases with increase in densities before reaching an asymptote, while in overcompensatory models recruitment declines as density increases (see [11, 14]). A thorough analysis of the dynamics of (4a) and (4b) with the Ricker function appears in . The results in [10, 11] clarify many issues with regard to the effects of harvesting in stage-structured models such as global convergence to 0 and the existence of a stable survival equilibrium as well as the so-called hydra effect for different harvesting scenarios and with different recruitment functions; this refers to the counter-intuitive situation where an increase in the harvest or mortality rate results in a corresponding increase in the total population; for example, see [15–17].
Also studied in  is the occurrence of periodic and nonperiodic attractors and chaotic behavior for certain parameter ranges.
Next, the model in  studies the harvesting and predation of sex- and age-structured populations. Although the added stage for two sexes results in a three-dimensional model, the existence of an attracting, invariant planar manifold reduces the study of the asymptotics of the system to that of the planar system:where the density-dependent per capita reproductive rate may be Beverton-Holt or Ricker similarly to in (4b). Here is the number of females and is the number of young members in the population (the male population is a fixed proportion of the females).
We also mention the adult-juvenile modelin which all adults are removed through harvesting, predation, migration, or just dying after one period, as in the case of semelparous species, that is, an organism that reproduces only once before death. In  conditions for the global attractivity of the positive fixed point and the occurrence of two cycles for (6a) and (6b) are obtained. A significant difference between (5a), (5b), (6a), and (6b) and systems (3a), (3b), (4a), and (4b) is the fact that in (5b) or in (6b) may depend on both and .
We study the qualitative properties of the orbits of (2a) and (2b) such as uniform boundedness and global convergence to 0 under minimal restrictions on time-dependent parameters. Biological constraints may be readily imposed to obtain special cases relevant to population models.
We also investigate convergence to zero with periodic parameters (extinction in a periodic environment). In particular, we show that convergence to zero occurs even if the mean value of exceeds 1, a situation that cannot occur if is constant in ; see Remark 16 below.
In the final section we study the dynamics of a rational special case of (2a) and (2b). Sufficient conditions for the global asymptotic stability of a fixed point in the positive quadrant as well as conditions for the occurrence of orbits of prime period two are obtained. In particular, we establish that a sufficiently high level of interspecies competition tends to destabilize the survival fixed point and result in periodic oscillations.
2. Uniform Boundedness of Orbits
Conditions under which the orbits of (2a) and (2b) are bounded are not transparent. In this section we obtain general results about the uniform boundedness of orbits of (2a) and (2b) in the positive quadrant . We begin with a simple, yet useful lemma.
Lemma 1. Let , let , and let . If for all then for every and all sufficiently large values of
Proof. Let and note that every solution of the linear, first-order equation converges to its fixed point . Furthermore, and, by induction, . Since for every and all sufficiently large ,
Theorem 2. Let , , be bounded on the compact sets in for each and suppose that for some that is, the sequence of functions is uniformly bounded on the square . If there are numbers and such that uniformly for all then all orbits of (2a) and (2b) are uniformly bounded and for all sufficiently large values of satisfy
Corollary 3. For functions , , defined on for assume that there are numbers and such that for all and all Then all orbits of (2a) and (2b) are uniformly bounded and for all sufficiently large values of
However, if , , and , then all orbits of this system with initial values in are uniformly bounded by Theorem 2.
3. Global Attractivity of the Origin
In this section we obtain general sufficient conditions for the convergence of all orbits of the system to . For population models these yield conditions that imply the extinction of species.
3.1. General Results
We start with the following lemma; see  for the proof and some background on this result.
Lemma 4. Let and assume that the functions satisfy the inequalityfor all and all . Then for every solution of the difference equationthe following is true:
Throughout this section we assume that , are all bounded functions for and every . Then the following are well-defined sequences of real numbers:
Theorem 5. If the inequalityholds, then for every orbit of the planar system (2a) and (2b) in the positive quadrant . If also either the sequence is bounded or the inequalityholds, then every orbit of (2a) and (2b) converges to .
Proof. By (26) there is such that for all (large) . From (2a)so for all (large) (2b) yieldsLemma 4 now implies that . Furthermore either by hypothesis there is a positive number such that or by (27) there is a positive number such that for all (large) so thatfor all sufficiently large values of Now, if or as the case may be, then from (2b) in the planar system we see thatand the proof is complete.
Remark 6. Theorem 5 is valid even if the separate sequences or are not bounded by 1 as long as for all large enough .
If (26) is satisfied but is unbounded and does not satisfy (27) then may not converge to 0; see the example following Corollary 18 below.
We consider an application of Theorem 5 to “noisy” autonomous system next. Let , , , be bounded sequences of real numbers and let
Additionally, let be bounded functions and denote their supremums over by , , , respectively. If in (2a) and (2b) we havethen we refer to (2a) and (2b) as an autonomous system with low-amplitude disturbances or fluctuations in the rates , , , assuming that all three of these are positive functions and for all
Corollary 7. Suppose that (2a) and (2b) is an autonomous system with low-amplitude disturbances or fluctuations in the above sense. Ifthen the origin is the unique, globally asymptotically stable fixed point of (2a) and (2b) relative to the positive quadrant .
Note that (35) holds for nontrivial sequences , of real numbers if .
Remark 8. Since in the above discussion the sequences , , , are arbitrary bounded sequences, they can also be sequences of random variables that are drawn from distributions with finite support. For example, , can be drawn from uniform distribution on some interval so long asCorollary 7 will hold, implying that the origin is globally attracting even in the presence of noise.
In the autonomous case where the three parameter functions , , do not depend on at all, we have the following planar system:
Corollary 9. Assume that are bounded functions and the following inequality holds:then the origin is the unique, globally asymptotically stable fixed point of (37a) and (37b) relative to the positive quadrant .
Inequality (38) may be explicitly related to the local asymptotic stability of the origin for (37a) and (37b) when the functions , , are smooth. Consider the associated mappingwhose linearization at has eigenvalues
These are real and a routine calculation shows that if
Under suitable differentiability hypotheses, this inequality is implied by (38) and is equivalent to it if the suprema of and occur at .
3.2. Folding the System
In the next and later sections it will be convenient to fold system (2a) and (2b) to a second-order equation; see  for more details on folding. System (2a) and (2b) in general folds as follows: substitute for from (2b) into (2a) to obtainwhereis derived by solving (2a) for . Although an explicit formula for is not feasible in general, it is readily obtained in typical cases; for instance, suppose that and are both independent of (or constant in) for all ; note that systems (3a), (3b), (4a), (4b), (5a), (5b), (6a), and (6b) are all of this type. In this case it is clear thatand furthermore (42) reduces to
The pair of first-order equations (44) and (45) represents folding of (2a) and (2b). Note that with positive parameter functions, each pair generates an orbit of (2a) and (2b) that is in for all . So we have and also by (43) so is well defined for every such orbit of (2a) and (2b).
Remark 10. An even simpler reduction than the above is possible if is independent of (or constant in) . In this case,and it is not necessary to solve (2a) for implicitly (i.e., the system folds without inversions). Special cases of this type include systems (3a), (3b), (4a), and (4b).
3.3. Global Convergence to Zero with Periodic Parameters
The right-hand side of the above inequality is a linear expression. Consider the linear difference equationwhere the coefficients and are nonnegative and their periods and are positive integers with least common multiple ; we say that the linear difference equation (48) is periodic with period . In this study we assume that
By Lemma 4 every solution of (48) converges to zero if for all . However, it is known that convergence to zero may occur even when exceeds 1 (for infinitely many in the periodic case). We use the approach in  to examine the consequences of this issue when the planar system has periodic parameters. The following result is an immediate consequence of Theorem in .
Lemma 11. Assume that , for are obtained by iteration from (48) from the real initial values:Suppose that the quadratic polynomialis proper, that is, not , and suppose that it has a real root . If the recurrencegenerates nonzero real numbers then is periodic with preiod and yields a triangular system of first-order equations that is equivalent to (48) as follows:
System (53) and (54) is also known as a semiconjugate factorization of (48); see  for an introduction to this concept. The sequence that is generated by (52) is said to be (unitary) eigensequence of (48). Eigenvalues are essentially constant eigensequences for if in Lemma 11 then (51) reduces to and the latter equation is the standard characteristic equation of (48) with constant coefficients; see  for more details on the semiconjugate factorization of linear difference equations.
Proof. (a) Let . Then and since by definition it follows that . Induction completes the proof that if . The converse is obvious since .
(b) Since and the stated inequalities hold for . If (58) is true for some then Now, the proof is completed by induction. The proof of (59) is similar sinceand if (59) holds for some then which establishes the induction step.
Lemma 13. Assume that (49) holds with for and (48) is periodic with period . Then one has the following.
(a) Equation (48) has a positive (hence unitary) eigensequence of period .
(b) If for thenHence, if(c) If for then .
Proof. (a) Lemma 12 shows that for . Now, either (i) or (ii) . In case (i), the root of the quadratic polynomial (51) is positive since by Lemma 12 and thusIf then from (52) for . Thus, by Lemma 11, (48) has a unitary (in fact positive) eigensequence of period . If then by Lemma 12 and (51) reduces towhich has a root . As in the previous case it follows that (48) has a positive eigensequence of period .
(b) To establish (63), let and note that (51) can be written asSince has period , so from (52) and the definition of the numbers and it follows that Since it follows thatWe claim that if for thenThis claim is easily seen to be true by induction; we showed that it is true for and if (70) holds for some then by (52) from which it follows thatand the induction argument is complete. Now, using (70), we obtainGiven that (73) implies that and (63) is obtained. Hence, ifUpon rearranging terms and squaring,which reduces to (64) after straightforward algebraic manipulations.
(c) First, assume that is odd. Then by (59)so from (63)If for then so as required. Now let be even. Then from (63) and (59)If for then and and the proof is complete.
Some of the numbers may exceed 1 in Lemma 13 without affecting the conclusions of the lemma. Additionally, not all the conditions in Lemma 13 are necessary. For instance, if then Lemma 13(c) holds trivially. Additionally, by Lemma 12(a), for so the following equality must hold instead of (63):
Lemma 14. Let and let for with . Then the linear equation (48) has a positive (hence unitary) eigensequence of period given byand if .
In Lemma 14 some of the numbers or may exceed 1 without affecting the conclusions of the lemma.
Theorem 15. Assume that (27) holds and the sequences and and have period with and for . Additionally let the numbers be as previously defined with and . All nonnegative orbits of the planar system converge to if either one of the following holds:(a) and (64) holds.(b) and .
Proof. Let be a solution of the linear equation (48) with , , , and . Then by (47) By induction it follows that . If (64) holds then, by Lemma 13, so converges to 0. Furthermore, as in the proof of Theorem 5 and the proof is complete.
Remark 16. Condition (64) involves the numbers , rather than the coefficients of (48) directly. In the case of period the role of and is more apparent. Inequality (64) in this case is and simple manipulations reduce the last inequality to Inequality (85) holds even if or thus showing how global convergence to my occur when (26) does not hold. Furthermore, it is possible that (85) holds together withNote that (85) holds even with arbitrarily large mean value in (86) if say as . In population models this implies that if (85) holds with and then extinction may still occur after restocking the adult population to raise the mean value of the composite parameter above 1 by a wide margin.
In Theorem 15 the individual sequences , need not be periodic or even bounded. Therefore, the theorem applies to (2a) and (2b) even if the system itself is not periodic as long as the combination of parameters is periodic along with .
4. Dynamics of a Beverton-Holt Type Rational System
In this section we apply some of the preceding results and obtain some new ones to study boundedness, extinction, and modes of survival in some rational special cases of (2a) and (2b). In population models these types of systems include the Beverton-Holt type interactions. Specifically, we consider the following nonautonomous system and some of its special cases:where we assume that for all and
For example, if we think of as the natural survival rates then the population model (3a) and (3b) is a special case of (87a) and (87b). If we allow to include additional factors such as harvesting rates then (87a) and (87b) is an extension of the model in  (with a Beverton-Holt recruitment function) in the sense that the competition coefficients , , and may be nonzero as well as time-dependent.
4.1. Uniform Boundedness and Extinction
Corollary 17. Assume that (88) holds.
(a) Let the sequence be bounded and . If there is such that for all larger than a given positive integer then all orbits of (87a) and (87b) are uniformly bounded.
(b) Let the sequence be bounded and suppose that there is such thatfor all larger than a given positive integer. Then all orbits of (87a) and (87b) are uniformly bounded.
Proof. (a) By hypothesis, for all (large) ,Next, letBy hypothesis, there is and such that for all and all sufficiently large values of Now an application of Corollary 3 completes the proof of (a).
(b) By (89) for all large it follows thatand, likewise,for all large . Therefore, . Next, if is bounded then is also bounded and the proof is complete.
The next result follows readily from Theorem 5.
The above corollary is false when (95) holds if is unbounded and thus has a subsequence that converges to 0.
Example 19. Consider system where , , , , , and . Then (95) is satisfied, so . But does not approach 0 for large enough ; this may be inferred from Lemma 4 which shows that converges to 0 at an exponential rate where . Thuswill not converge to 0 if is sufficiently large.
It is noteworthy that if in part (a) above then (98a) and (98b) may have unbounded solutions as in, for example, system where and the remaining parameters are positive. This system folds to the second-order rational equationwhich is known to have unbounded solutions if ; see .
Corollary 18 likewise simplifies in the autonomous case.
4.2. Persistence and the Role of Competition
We now explore the effects of competition in the autonomous system (98a) and (98b). There are 6 different competition coefficients and to reduce the number of different cases we focus on the special case below where :
By the last two corollaries, all orbits of the rational system (102) and (103) in are uniformly bounded if and and they converge to the origin if . We now examine this rational system in more detail using its folding, namely, the second-order rational equationwhere
With initial values and derived from , -component of the orbits of the system is obtained by iterating (104). The equation in (106) is passive in the sense that after -component of the orbit is generated by the core equation (104) -component is derived from (106) without any further iterations. This observation also establishes the nontrivial fact that solutions of (104) that correspond to the orbits of the system in are nonnegative and well-defined even for .
If , that is, , then zero is the only fixed point of (104). Corollary 21 establishes that, in this case, zero is globally asymptotically stable relative to . On the other hand, when , that is, , then is no longer a stable fixed point of (104). By routine calculations, one can show that zero is a saddle point when and if then zero is a repeller.
We note that is also a positive fixed point of folding (104). Under certain conditions, attracts all solutions of (104) with positive initial values, and it is thus a survival equilibrium. We state the following result from literature; see .
Lemma 22. Let be an open interval of real numbers and suppose that is nondecreasing in each coordinate. Let be a fixed point of the difference equationand assume that the function satisfies the conditionsThen is an invariant interval of (108) and attracts all solutions with initial values in
Theorem 23. Let ; that is, . If the functionis nondecreasing in both arguments, then the fixed point attracts all solutions of (104) with initial values in .
Proof. If we letthen the fixed point is the solution of . For , we may write whereNow,for all , so is strictly decreasing for all . Therefore, The rest of the proof follows from Lemma 22.
If and then , so . Therefore by Theorem 23 is globally asymptotically stable. However, if , then may not be positive, so the results of Theorem 23 may not apply to this case. The next result shows that orbits of the system may converge to if but not too large.
Theorem 24. Let and let ; that is, . Then there exists such that for the fixed point of (104) is globally asymptotically stable relative to .
Proof. Sinceto ensure that it suffices for ; that is,which is equivalent toand the proof is complete.
If is sufficiently large then is not positive on . Furthermore, also becomes unstable for large enough , which we establish next by examining the linearization of (104) around .
The characteristic equation associated with the linearization of (104) at is given bywhere
The roots of (119) are given by
Since for all it follows that and both roots are real with and . Furthermore, ifwhich is equivalent to
This inequality holds, since under our assumptions on the parameters. Therefore, . On the other hand, if and only ifwhich is equivalent to
Note that when this is trivially the case since under our assumptions on the parameters. Thus, is locally asymptotically stable if .
Next, if and